Introduction to the Theory of Infinite-Dimensional Dissipative Systems

I. D. Chueshov Author: I. D. Chueshov This book provides an exhaustive introduction to the scope Introduction to the Theory Title: Introduction to the Theory of InfiniteDimensional of InfiniteDimensional of main ideas and methods of the Dissipative Systems Dissipative Systems theory of infinite-dimensional dissipative dynamical systems which 966–7021–64–5 ISBN: 966–7021–64–5 has been rapidly developing in recent years. In the examples sys tems generated by nonlinear partial differential equations I.D.Chueshov I . D . Chueshov arising in the different problems of modern mechanics of continua are considered. The main goal of the book is to help the reader Introduction Theory Introduction to the Theory to master the basic strategies used Infinite-Dimensional of Infinite-Dimensional in the study of infinite-dimensional dissipative systems and to qualify Dissipative Dissipative him/her for an independent scien-

Systems 2002 Systems tific research in the given branch. Experts in nonlinear dynamics will find many fundamental facts in the convenient and practical form » in this book. Universitylecturesincontemporarymathematics The core of the book is composed of the courses given by the author at the Department of Me chanics and Mathematics at Kharkov University during a number of years. This book contains a large number of exercises which make the main text more complete. It is sufficient to know the fundamentals of functional analysis and ordinary differential equations to read the book. CTA Translated by Constantin I. Chueshov ORDER You can ORDER this book from the Russian edition («ACTA», 1999) while visiting the website of «ACTA» Scientific Publishing House Translation edited by www.acta.com.ua http://www.acta.com.uawww.acta.com.ua/en/ Maryna B. Khorolska A « I. D. Chueshov

A CTA 2002 UDC 517 2000 Mathematics Subject Classification: primary 37L05; secondary 37L30, 37L25.

This book provides an exhaustive introduction to the scope of main ideas and methods of the theory of infinite-dimensional dissipative dynamical systems which has been rapidly developing in recent years. In the examples systems generated by nonlinear partial differential equations arising in the different problems of modern mechanics of continua are considered. The main goal of the book is to help the reader to master the basic strategies used in the study of infinite-dimensional dissipative systems and to qualify him/her for an independent scientific research in the given branch. Experts in nonlinear dynamics will find many fundamental facts in the convenient and practical form in this book. The core of the book is composed of the courses given by the author at the Department of Mechanics and Mathematics at Kharkov University during a number of years. This book contains a large number of exercises which make the main text more complete. It is sufficient to know the fundamentals of functional analysis and ordinary differential equations to read the book.

Translated by Constantin I. Chueshov from the Russian edition («ACTA», 1999) Translation edited by Maryna B. Khorolska

ACTA Scientific Publishing House Kharkiv, Ukraine E-mail: [email protected] Свідоцтво ДК №179 .acta.com.ua © I. D. Chueshov, 1999, 2002 © Series, «ACTA», 1999

www © Typography, layout, «ACTA», 2002

ISBN 966-7021-20-3 (series) ISBN 966-7021-64-5 Contents

. . . . Preface...... 7

Chapter 1 . Basic Concepts of ththee Theory of Infinite-Dimensional Dynamical SySystststemsems

. . . . § 1 Notion of Dynamical System ...... 11 . . . . § 2 Trajectories and Invariant Sets ...... 17 . . . . § 3 Definition of Attractor ...... 20 . . . . § 4 Dissipativity and Asymptotic Compactness ...... 24 . . . . § 5 Theorems on Existence of Global Attractor ...... 28 . . . . § 6 On the Structure of Global Attractor ...... 34 . . . . § 7 Stability Properties of Attractor and Reduction Principle . . 45 . . . . § 8 Finite Dimensionality of Invariant Sets ...... 52 . . . . § 9 Existence and Properties of Attractors of a Class of Infinite-Dimensional Dissipative Systems ...... 61 . . . . References ...... 73

Chapter 2 . Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

. . . . § 1 Positive Operators with Discrete Spectrum ...... 77 . . . . § 2 Semilinear Parabolic Equations in Hilbert Space ...... 85 . . . . § 3 Examples ...... 93 . . . . § 4 Existence Conditions and Properties of Global Attractor . . 101 . . . . § 5 Systems with Lyapunov Function ...... 108 . . . . § 6 Explicitly Solvable Model of Nonlinear Diffusion ...... 118 . . . . § 7 Simplified Model of Appearance of Turbulence in Fluid . . . 130 . . . . § 8 On Retarded Semilinear Parabolic Equations ...... 138 . . . . References ...... 145 4 Contents

Chapter 3 . Inertial Manifolds

. . . . § 1 Basic Equation and Concept of Inertial Manifold ...... 149 . . . . § 2 Integral Equation for Determination of Inertial Manifold . . 155 . . . . § 3 Existence and Properties of Inertial Manifolds ...... 161 . . . . § 4 Continuous Dependence of Inertial Manifold on Problem Parameters ...... 171 . . . . § 5 Examples and Discussion ...... 176 . . . . § 6 Approximate Inertial Manifolds for Semilinear Parabolic Equations ...... 182 . . . . § 7 Inertial Manifold for Second Order in Time Equations . . . . 189 . . . . § 8 Approximate Inertial Manifolds for Second Order in Time Equations ...... 200 . . . . § 9 Idea of Nonlinear Galerkin Method ...... 209 . . . . References ...... 214

Chapter 4 . The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

. . . . § 1 Spaces ...... 218 . . . . § 2 Auxiliary Linear Problem ...... 222 . . . . § 3 Theorem on the Existence and Uniqueness of Solutions . . 232 . . . . § 4 Smoothness of Solutions ...... 240 . . . . § 5 Dissipativity and Asymptotic Compactness ...... 246 . . . . § 6 Global Attractor and Inertial Sets ...... 254 . . . . § 7 Conditions of Regularity of Attractor ...... 261 . . . . § 8 On Singular Limit in the Problem of Oscillations of a Plate ...... 268 . . . . § 9 On Inertial and Approximate Inertial Manifolds ...... 276 . . . . References ...... 281 Contents 5

Chapter 5 . Theory of FunFunctctctionalsionals ththatat Uniquely Determine Long-Time Dynamics

. . . . § 1 Concept of a Set of Determining Functionals ...... 285 . . . . § 2 Completeness Defect ...... 296 . . . . § 3 Estimates of Completeness Defect in Sobolev Spaces . . . . 306 . . . . § 4 Determining Functionals for Abstract Semilinear Parabolic Equations ...... 317 . . . . § 5 Determining Functionals for Reaction-Diffusion Systems . . 328 . . . . § 6 Determining Functionals in the Problem of Nerve Impulse Transmission ...... 339 . . . . § 7 Determining Functionals for Second Order in Time Equations ...... 350 . . . . § 8 On Boundary Determining Functionals ...... 358 . . . . References ...... 361

Chapter 6 . Homoclinic Chaos in Infinite-Dimensional SySystststemsems

. . . . § 1 Bernoulli Shift as a Model of Chaos ...... 365 . . . . § 2 Exponential Dichotomy and Difference Equations ...... 369 . . . . § 3 Hyperbolicity of Invariant Sets for Differentiable Mappings ...... 377 . . . . § 4 Anosov’s Lemma on A -trajectories ...... 381 . . . . § 5 Birkhoff-Smale Theorem ...... 390 . . . . § 6 Possibility of Chaos in the Problem of Nonlinear Oscillations of a Plate ...... 396 . . . . § 7 On the Existence of Transversal Homoclinic Trajectories . . 402 . . . . References ...... 413

. . . . Index ...... 415 Палкой щупая дорогу, Бродит наугад слепой, Осторожно ставит ногу И бормочет сам с собой. И на бельмах у слепого Полный мир отображен: Дом, лужок, забор, корова, Клочья неба голубого — Все, чего не видит он.

Вл. Ходасевич «Слепой»

A blind man tramps at random touching the road with a stick. He places his foot carefully and mumbles to himself. The whole world is displayed in his dead eyes. There are a house, a lawn, a fence, a cow and scraps of the blue sky — everything he cannot see.

Vl. Khodasevich «A Blind Man» Preface

The recent years have been marked out by an evergrowing interest in the research of qualitative behaviour of solutions to nonlinear evolutionary partial differential equations. Such equations mostly arise as mathematical models of processes that take place in real (physical, chemical, biological, etc.) systems whose states can be characterized by an infinite number of parameters in general. Dissipative systems form an important class of systems observed in reality. Their main feature is the presence of mechanisms of energy reallocation and dissipation. Interaction of these two mechanisms can lead to appearance of complicated limit regimes and structures in the system. Intense interest to the infinite-dimensional dissipative systems was significantly stimulated by attempts to find adequate mathematical models for the explanation of turbulence in liquids based on the notion of strange (irregular) attractor. By now significant progress in the study of dynamics of infinite-dimensional dissipative systems have been made. Moreover, the latest mathematical studies offer a more or less common line (strategy), which when followed can help to answer a number of principal questions about the properties of limit regimes arising in the system under consideration. Although the methods, ideas and concepts from finite-dimensional dynamical systems constitute the main source of this strategy, finite-dimensional approaches require serious revaluation and adaptation.

The book is devoted to a systematic introduction to the scope of main ideas, methods and problems of the mathematical theory of infinite-dimensional dissipative dynamical systems. Main attention is paid to the systems that are generated by nonlinear partial differential equations arising in the modern mechanics of continua. The main goal of the book is to help the reader to master the basic strategies of the theory and to qualify him/her for an independent scientific research in the given branch. We also hope that experts in nonlinear dynamics will find the form many fundamental facts are presented in convenient and practical.

The core of the book is composed of the courses given by the author at the Department of Mechanics and Mathematics at Kharkov University during several years. The book consists of 6 chapters. Each chapter corresponds to a term course (34-36 hours) approximately. Its body can be inferred from the table of contents. Every chapter includes a separate list of references. The references do not claim to be full. The lists consist of the publications referred to in this book and offer additional works recommen- 8 Preface

ded for further reading. There are a lot of exercises in the book. They play a double role. On the one hand, proofs of some statements are presented as (or contain) cycles of exercises. On the other hand, some exercises contain an additional information on the object under consideration. We recommend that the exercises should be read at least. Formulae and statements have double indexing in each chapter (the first digit is a section number). When formulae and statements from another chapter are referred to, the number of the corresponding chapter is placed first.

It is sufficient to know the basic concepts and facts from functional analysis and ordinary differential equations to read the book. It is quite un- derstandable for under-graduate students in Mathematics and Physics.

I.D. Chueshov Chapter 1

Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

Contents

. . . . § 1 Notion of Dynamical System ...... 11 . . . . § 2 Trajectories and Invariant Sets ...... 17 . . . . § 3 Definition of Attractor ...... 20 . . . . § 4 Dissipativity and Asymptotic Compactness ...... 24 . . . . § 5 Theorems on Existence of Global Attractor ...... 28 . . . . § 6 On the Structure of Global Attractor ...... 34 . . . . § 7 Stability Properties of Attractor and Reduction Principle . . . 45 . . . . § 8 Finite Dimensionality of Invariant Sets ...... 52 . . . . § 9 Existence and Properties of Attractors of a Class of Infinite-Dimensional Dissipative Systems ...... 61

. . . . References ...... 73

The mathematical theory of dynamical systems is based on the qualitative theory of ordinary differential equations the foundations of which were laid by Henri Poincaré (1854–1912). An essential role in its development was also played by the works of A. M. Lyapunov (1857–1918) and A. A. Andronov (1901–1952). At present the theory of dynamical systems is an intensively developing branch of mathematics which is closely connected to the theory of differential equations. In this chapter we present some ideas and approaches of the theory of dynamical systems which are of general-purpose use and applicable to the systems generated by nonlinear partial differential equations.

§ 1 Notion of Dynamical System

(), In this book dynamical system is taken to mean the pair of objects XSt consisting of a complete metric space XS and a family t of continuous mappings of the space X into itself with the properties = t Î = St + t St ° St , t, T+ , S0 I , (1.1) where T+ coincides with either a set R+ of nonnegative real numbers or a set = {},,,¼ = ()= Z+ 012 . If T+ R+ , we also assume that yt St y is a continuous function with respect to tyX for any Î . Therewith X is called a phase space, or a state space, the family St is called an evolutionary operator (or semigroup), parameter t Î T+ plays the role of time. If T+ = Z+ , then dynamical system is = (), called discrete (or a system with discrete time). If T+ R+ , then XSt is frequently called to be dynamical system with continuous time. If a notion of dimension can be defined for the phase space XX (e. g., if is a lineal), the value dim X is called a dimension of dynamical system. Originally a dynamical system was understood as an isolated mechanical system the motion of which is described by the Newtonian differential equations and which is characterized by a finite set of generalized coordinates and velocities. Now people associate any time-dependent process with the notion of dynamical system. These processes can be of quite different origins. Dynamical systems naturally arise in physics, chemistry, biology, economics and sociology. The notion of dynamical system is the key and uniting element in synergetics. Its usage enables us to cover a rather wide spectrum of problems arising in particular sciences and to work out universal approaches to the description of qualitative picture of real phenomena in the universe. 12 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C h Let us look at the following examples of dynamical systems. a p t Example 1.1 e r Let fx() be a continuously differentiable function on the real axis posessing the 1 property xf() x ³ -C ()1 + x2 , where C is a constant. Consider the Cauchy problem for an ordinary differential equation \ ()= - ()() > ()= x t fxt , t 0 ,x 0 x0 . (1.2) For any x Î R problem (1.2) is uniquely solvable and determines a dynamical = () system in R . The evolutionary operator St is given by the formula St x0 xt , where xt() is a solution to problem (1.2). Semigroup property (1.1) holds by virtue of the theorem of uniqueness of solutions to problem (1.2). Equations of the type (1.2) are often used in the modeling of some ecological processes. For example, if we take fx()=a a× xx()- 1 , > 0 , then we get a logistic equation that describes a growth of a population with competition (the value xt() is the population level; we should take R+ for the phase space).

Example 1.2 Let fx() and gx() be continuously differentiable functions such that x Fx()= ò f ()xx d ³ -c , gx()³ -c 0 with some constant c . Let us consider the Cauchy problem \\ \ ì x ++gx()x fx()= 0 , t> 0 , í (1.3) ()= \ ()= î x 0 x0 , x 0 x1 . = (), Î 2 For any y0 x0 x1 R , problem (1.3) is uniquely solvable. It generates ()2, a two-dimensional dynamical system R St , provided the evolutionary operator is defined by the formula (); = ()() \ () St x0 x1 xt; x t , where xt() is the solution to problem (1.3). It should be noted that equations of the type (1.3) are known as Liénard equations in literature. The van der Pol equation: gx()== e()x2 - 1 , e > 0 , fx() x and the Duffing equation: gx()== e, e > 0 , fx() x3 - ax× - b which often occur in applications, belong to this class of equations. Notion of Dynamical System 13

Example 1.3 Let us now consider an autonomous system of ordinary differential equations \ ()= (),,,¼ = ,,¼ , xk t fk x1 x2 xN , k 12 N .(1.4) Let the Cauchy problem for the system of equations (1.4) be uniquely solvable over an arbitrary time interval for any initial condition. Assume that a solution continuously depends on the initial data. Then equations (1.4) generate an N - di- ()N, mensional dynamical system R St with the evolutionary operator St acting in accordance with the formula ==()(),, ¼ () (),,,¼ St y0 x1 t xN t , y0 x10 x20 xN 0 , {}() where xi t is the solution to the system of equations (1.4) such that ()= = ,,¼ , xi 0 xi 0 , i 12 N . Generally, let XF be a linear space and be a continuous mapping of X into itself. Then the Cauchy problem \ ()==()() > () Î x t Fxt , t 0 , x 0 x0 X (1.5) (), generates a dynamical system XSt in a natural way provided this problem is well-posed, i.e. theorems on existence, uniqueness and continuous dependence of solutions on the initial conditions are valid for (1.5).

Example 1.4 Let us consider an ordinary retarded differential equation \ x ()t + axt()= fxt()()- 1 , t > 0 , (1.6) where f is a continuous function on R1 , a > 0 . Obviously an initial condition for (1.6) should be given in the form xt() = f()t .(1.7) t Î []-10, Assume that f ()t lies in the space C []-1, 0 of continuous functions on the segment []-1, 0 . In this case the solution to problem (1.6) and (1.7) can be constructed by step-by-step integration. For example, if 0 ££t 1, the solution xt() is given by t -a ()- t xt()= e-a t f ()0 + òe t f ()tft()-1 d , 0 and if t Î []12, , then the solution is expressed by the similar formula in terms of the values of the function xt() for t Î []01, and so on. It is clear that the solution is uniquely determined by the initial function f ()t . If we now define an = []- , operator St in the space XC1 0 by the formula ()tf ()= ()+ t t Î []- , St xt , 10, where xt() is the solution to problem (1.6) and (1.7), then we obtain an infi- ()[]- , , nite-dimensional dynamical system C 1 0 St . 14 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C h Now we give several examples of discrete dynamical systems. First of all it should be a (), noted that any system XSt with continuous time generates a discrete system if p Î Î t we take t Z+ instead of t R+ . Furthermore, the evolutionary operator St of e a discrete dynamical system is a degree of the mapping S , i. e. S = St ,tÎ . r 1 t 1 Z+ Thus, a dynamical system with discrete time is determined by a continuous mapping 1 of the phase space X into itself. Moreover, a discrete dynamical system is very often defined as a pair ()XS, , consisting of the metric space X and the continuous mapping S .

Example 1.5 Let us consider a one-step difference scheme for problem (1.5): x - x n +1 n = () =t,,,¼ > t Fxn , n 012 , 0 . There arises a discrete dynamical system ()XS, n , where S is the continuous mapping of XSxx into itself defined by the formula = + t Fx() .

Example 1.6 Let us consider a nonautonomous ordinary differential equation \ x ()t = fx(), t, t > 0 ,x Î R1 , (1.9) where fx(), t is a continuously differentiable function of its variables and is periodic with respect to t , i. e. fx(), t = fx(), t+ T for some T > 0 . It is assumed that the Cauchy problem for (1.9) is uniquely solvable on any time interval. We define a monodromy operator (a period mapping) by the formula = () () Sx0 xT, where xt is the solution to (1.9) satisfying the initial condition ()= x 0 x0 . It is obvious that this operator possesses the property Sk xt()= xt()+ kT (1.10)

for any solution xt() to equation (1.9) and anyk Î Z+ . The arising dynamical system ()R1, Sk plays an important role in the study of the long-time properties of solutions to problem (1.9).

Example 1.7 (Bernoulli shift) = S = {}, Î Let X 2 be a set of sequences xxi i Z consisting of zeroes and ones. Let us make this set into a metric space by defining the distance by the formula (), = { -n = ,}< dx y inf 2 : xi yi in. Let SX be the shift operator on , i. e. the mapping transforming the sequence = {} = {} = xxi into the element yyi , where yi xi +1 . As a result, a dynamical system ()XS, n comes into being. It is used for describing complicated (qua- sirandom) behaviour in some quite realistic systems. Notion of Dynamical System 15

In the example below we describe one of the approaches that enables us to connect dynamical systems to nonautonomous (and nonperiodic) ordinary differential equations.

Example 1.8 Let hx(), t be a continuous bounded function on R2 . Let us define the hull (), Lh of the function hx t as the closure of a set ìü íýht ()xt, º hx(), t+t , t Î R îþ with respect to the norm ìü h = sup íýhx(), t : x Î R, t Î R . C îþ Let gx() be a continuous function. It is assumed that the Cauchy problem

\ ()==()+ (), () x t gx h xt, x 0 x0 (1.11)

[ , ¥) Î is uniquely solvable over the interval 0+ for any h Lh . Let us define the evolutionary operator S on the space X = R1 ´ L by the formula t h (), = ()()t , St x0 h x ht ,

where xt() is the solution to problem (1.11) and ht = h ()xt, +t . As a result, ()´ , a dynamical system R Lh St comes into being. A similar construction is often used when Lh is a compact set in the space C of continuous bounded functions (for example, if hx(), t is a quasiperiodic or almost periodic function). As the following example shows, this approach also enables us to use naturally the notion of the dynamical system for the description of the evolution of objects subjected to random influences.

Example 1.9

Assume that f0 and f1 are continuous mappings from a metric space Y into itself. Let Yy be a state space of a system that evolves as follows: if is the state of + () () the system at time kk , then its state at time 1 is either f0 y or f1 y with ¤ probability 12 , where the choice of f0 or f1 does not depend on time and the previous states. The state of the system can be defined after a number of steps in time if we flip a coin and write down the sequence of events from the right to the left using 01 and . For example, let us assume that after 8 flips we get the following set of outcomes: ¼ 10 110010 . Here 10 corresponds to the head falling, whereas corresponds to the tail falling. Therewith the state of the system at time t = 8 will be written in the form: 16 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C = ()() h Wf1 ° f0 ° f1 ° f1 ° f0 ° f0 ° f1 ° f0 y . a S p This construction can be formalized as follows. Let 2 be a set of two-sided se- t e quences consisting of zeroes and ones (as in Example 1.7), i.e. a collection r of elements of the type w¼w= ()¼w w w ¼w ¼ 1 -n -1 0 1 n , w = S ´ where i is equal to either 10 or . Let us consider the space X 2 Y con- =wS()w, Î Î sisting of pairs x y , where 2 , yY . Let us define the mapping FX:® X by the formula:

Fx()º F()w, y = ()S w, fw ()y , 0 S where S is the left-shift operator in 2 (see Example 1.7). It is easy to see that the n - th degree of the mapping F actcts according to the formula n n F ()w, y = ()S w , ()fw ° ¼ ° fw ° fw ()y n - 1 1 0 ()S ´ , n and it generates a discrete dynamical system 2 Y F . This system is often called a universal random (discrete) dynamical system.

Examples of dynamical systems generated by partial differential equations will be given in the chapters to follow.

Exercise 1.1 Assume that operators S have a continuous inverse for any t . t Show that the family of operators {}S : t Î R defined by the equa- t = ³ = -1 < lity St St for t 0 and St S t for t 0 form a group, i.e. (1.1) holds for all t, t Î R .

Exercise 1.2 Prove the unique solvability of problems (1.2) and (1.3) involved in Examples 1.1 and 1.2.

Exercise 1.3 Ground formula (1.10) in Example 1.6.

Exercise 1.4 Show that the mapping St in Example 1.8 possesses semigroup property (1.1).

Exercise 1.5 Show that the value dx(), y involved in Example 1.7 is a metric. Prove its equivalence to the metric ¥ *(), = - i - d xy å 2 xi yi . i = -¥ Trajectories and Invariant Sets 17

§ 2 Trajectories and Invariant Sets

(), Let XSt be a dynamical system with continuous or discrete time. Its trajectory (or orbit) is defined as a set of the type g = {}ut(): t Î T , where ut() is a continuous function with values in XS such that t ut()= ut()+ t for all t Î T+ and t Î T . Positive (negative) semitrajectory is defined as a set + – g =g{}ut(): t ³ 0 , (= {}ut(): t £ 0 , respectively), where a continuous on T+ () ()= ()+t (T– , respectively) function ut possesses the property St ut ut for any t > 0 , t ³t0 (> 0, t £ 0, t + t £ 0 , respectively). It is clear that any positive g+ g+ = {}³ semitrajectory has the form St v : t 0 , i.e. it is uniquely determined by its initial state v . To emphasize this circumstance, we often write g+ = g+()v . In general, it is impossible to continue this semitrajectory g+()v to a full trajectory without imposing any additional conditions on the dynamical system.

Exercise 2.1 Assume that an evolutionary operator St is invertible for some t > 0 . Then it is invertible for all t > 0 and for any vXÎ there exists a negative semitrajectory g– = g–()v ending at the point v .

A trajectory g = {}ut(): t Î T is called a periodic trajectory (or a cycle) if there exists T Î T+ ,T > 0 such that ut()+ T = ut() . Therewith the minimal number T > 0 possessing the property mentioned above is called a period of a trajectory. Here TRZ is either or depending on whether the system is a continuous Î or a discrete one. An element u0 X is called a fixed point of a dynamical system (), = ³ XSt if St u0 u0 for all t 0 (synonyms: equilibrium point, stationary point).

(), Exercise 2.2 Find all the fixed points of the dynamical system RSt generated by equation (1.2) with fx()= xx()- 1 . Does there exist a periodic trajectory of this system?

Exercise 2.3 Find all the fixed points and periodic trajectories of a dynamical system in R2 generated by the equations

ì \ = - a - []()2 + 2 2 - ()2 + 2 + ïx y xx y 4 x y 1 , í ï \ = a - []()2 + 2 2 - ()2 + 2 + îy xyx y 4 x y 1 . Consider the cases a ¹a0 and = 0 . Hint: use polar coordinates.

Exercise 2.4 Prove the existence of stationary points and periodic trajectories of any period for the discrete dynamical system described 18 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C h in Example 1.7. Show that the set of all periodic trajectories is dense a in the phase space of this system. Make sure that there exists a tra- p t jectory that passes at a whatever small distance from any point of the e phase space. r

1 The notion of invariant set plays an important role in the theory of dynamical systems. A subset YX of the phase space is said to be: Í ³ a) positively invariant, if St YY for all t 0 ; Ê ³ b) negatively invariant, if St YY for all t 0 ; c) invariant, if it is both positively and negatively invariant, i.e. if = ³ St YY for all t 0 . The simplest examples of invariant sets are trajectories and semitrajectories.

Exercise 2.5 Show that g+ is positively invariant, g– is negatively invariant and g is invariant.

Exercise 2.6 Let us define the sets g+()= ()º {}= Î A È St A È vSt u : uA t ³ 0 t ³ 0 and g–()= -1 ()º {}Î A È St A È v : St vA t ³ 0 t ³ 0 for any subset AX of the phase space . Prove that g+()A is a positively > invariant set, and if the operator St is invertible for some t 0, then g–()A is a negatively invariant set.

Other important example of invariant set is connected with the notions of w -limit and a -limit sets that play an essential role in the study of the long-time behaviour of dynamical systems. Let AXÌw . Then the -limit set for A is defined by w ()= () A Ç È St A , s ³ 0 ts³ X ()= {}= Î [] where St A vSt u : uA . Hereinafter Y X is the closure of a set Y in the space X . The set a ()= -1 () A Ç È St A , s ³ 0 ts³ X -1 ()=a{}Î where St A v : St vA , is called the -limit set for A . Trajectories and Invariant Sets 19

Lemma 2.1 For an element y to belong to an ww -limit set ()A , it is necessary and {}Ì sufficient that there exist a sequence of elements yn A and a sequence of numbers tn , the latter tending to infinity such that lim dS()y , y = , n ® ¥ tn n where dx(), y is the distance between the elements xy and in the space X .

Proof. Let the sequences mentioned above exist. Then it is obvious that for any t > ³ 0 there exists n0 0 such that S y Î S ()A , nn³ . tn n È t 0 t ³ t This implies that = Î () ySlim t yn È St A n ® ¥ n t ³ t X for all t > 0 . Hence, the element y belongs to the intersection of these sets, i.e. y Î w ()A . On the contrary, if y Î w ()A , then for all n = 012,,,¼ Î () ySÈ t A . tn³ X

Hence, for any nz there exists an element n such that Î () (), £ 1 zn È St A , dy zn n . tn³ Therewith it is obvious that z = S y ,y Î A ,t ³ n . This proves the n tn n n n lemma.

It should be noted that this lemma gives us a description of an w -limit set but does not guarantee its nonemptiness.

Exercise 2.7 Show that w ()A is a positively invariant set. If for any t > 0 w () there exists a continuous inverse to St , then A is invariant, i.e. w ()= w() St A A . > Exercise 2.8 Let St be an invertible mapping for every t 0 . Prove the counterpart of Lemma 2.1 for an a -limit set: ìü Î a ()Û ${}Î , $ , ® ¥ ()-1 , = y A íýyn A tn tn + ; lim dSt yn y 0 . îþn ® ¥ n Establish the invariance of a ()A . 20 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C h Exercise 2.9 Let g = {}ut(): -¥ < 0, r > 0, yXÎ dt t t Then the set {}y : Vy()£ R is positively invariant for any R ³ ³ ra¤ .

§ 3 Definition of AAttttractortractor

Attractor is a central object in the study of the limit regimes of dynamical systems. Several definitions of this notion are available. Some of them are given below. From the point of view of infinite-dimensional systems the most convenient concept is that of the global attractor. Ì A bounded closed set A1 X is called a global attractor for a dynamical sys- (), tem XSt , if = > 1) A1 is an invariant set, i.e. St A1 A1 for any t 0 ; 2) the set A1 uniformly attracts all trajectories starting in bounded sets, i.e. for any bounded set BX from

ìü (), Î = limsup íýdistSt yA1 : yB 0 . t ® ¥ îþ We remind that the distance between an element zA and a set is defined by the equality: dist ()zA, = inf {}dz(), y: yAÎ , where dz(), y is the distance between the elements zyX and in . The notion of a weak global attractor is useful for the study of dynamical systems generated by partial differential equations. Definition of Attractor 21

Let XA be a complete linear metric space. A bounded weakly closed set 2 is ()= , > called a global weak attractor if it is invariant St A2 A2 t 0 and for any O Ì weak vicinity of the set A2 and for every bounded set BX there exists = ()O, Ì O ³ t0 t0 B such that St B for tt0 . We remind that an open set in weak topology of the space X can be described as finite intersection and subsequent arbitrary union of sets of the form = {}Î ()< Ulc, xX: lx c , where cl is a real number and is a continuous linear functional on X . It is clear that the concepts of global and global weak attractors coincide in the finite-dimensional case. In general, a global attractor A is also a global weak attractor, provided the set A is weakly closed.

Exercise 3.1 Let A be a global or global weak attractor of a dynamical sys- (), tem XSt . Then it is uniquely determined and contains any bounded negatively invariant set. The attractor A also contains the w - limit set w ()B of any bounded BXÌ . (), Exercise 3.2 Assume that a dynamical system XSt with continuous time possesses a global attractor A1 . Let us consider a discrete system ()XT, n , where TS= with some t > 0 . Prove that A is a glo- t0 0 1 bal attractor for the system ()XT, n . Give an example which shows that the converse assertion does not hold in general.

If the global attractor A1 exists, then it contains a global minimal attractor A3 which is defined as a minimal closed positively invariant set possessing the property (), = Î limdist St yA3 0 for everyyX . t ® ¥

By definition minimality means that A3 has no proper subset possessing the properties mentioned above. It should be noted that in contrast with the definition of the global attractor the uniform convergence of trajectories to A3 is not expected here. = Exercise 3.3 Show that St A3 A3 , provided A3 is a compact set. w ()Î Î Exercise 3.4 Prove that x A3 for any xX . Therewith, if A3 is = {}w () Î a compact, then A3 È x : xX .

By definition the attractor A3 contains limit regimes of each individual trajectory. ¹ It will be shown below that A3 A1 in general. Thus, a set of real limit regimes (states) originating in a dynamical system can appear to be narrower than the global attractor. Moreover, in some cases some of the states that are unessential from the point of view of the frequency of their appearance can also be “removed” from A3 , for example, such states like absolutely unstable stationary points. The next two definitions take into account the fact mentioned above. Unfortunately, they require 22 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C h additional assumptions on the properties of the phase space. Therefore, these defini- a tions are mostly used in the case of finite-dimensional dynamical systems. p t Let a Borel measure mm such that ()X < ¥ be given on the phase space X of e a dynamical system ()XS, . A bounded set A in X is called a Milnor attractor r t 4 Milnor attractor m (), (with respect to the measure ) for XSt if A4 is a minimal closed invariant set 1 possessing the property (), = limdist St yA4 0 t ® ¥ for almost all elements yXÎm with respect to the measure . The Milnor attractor is frequently called a probabilistic global minimal attractor. At last let us introduce the notion of a statistically essential global minimal at- () tractor suggested by Ilyashenko. Let UX be an open set in X and let U x be its ()= Î ()= Ï characteristic function: XU x 1 , xU ; XU x 0 , xU . Let us define the average time t ()xU, which is spent by the semitrajectory g+()x emanating from x in the set U by the formula T t (), = 1 () xU lim XU St x dt . T ® ¥ T ò 0 A set U is said to be unessential with respect to the measure m if MU()º m{}x : t ()xU, > 0 = 0 .

The complement A5 to the maximal unessential open set is called an Ilyashenko aattttractortractor (with respect to the measure m ).

It should be noted that the attractors A4 and A5 are used in cases when the natural Borel measure is given on the phase space (for example, if X is a closed measurable set in RN and m is the Lebesgue measure). The relations between the notions introduced above can be illustrated by the following example.

Example 3.1 Let us consider a quasi-Hamiltonian system of equations in R2 :

ì \ ¶H ¶H ïq = - m H , ï ¶p ¶q í (3.1) ï \ ¶H ¶H ïp = -m - H , î ¶q ¶p where Hp(), q =m()12¤ p2 + q4 - q2 and is a positive number. It is easy to ascertain that the phase portrait of the dynamical system generated by equations (3.1) has the form represented on Fig. 1. Definition of Attractor 23

A separatrix (“eight curve”) separates the domains of the phase plane with the different qualitative behaviour of the trajectories. It is given by the equation Hpq(), = 0 . The points ()pq, inside the separatrix are characterized by the equation Hpq(), < 0 . Therewith it appears that

Fig. 1. Phase portrait of system (3.1)

=={}(), (), £ A1 A2 pq: Hp q 0 , ìüìü¶ ¶ A = íý()pq, : Hp(), q= 0 È íý()pq, : Hp(), q==Hp(), q 0 , 3 îþîþ¶p ¶q = {}(), (), = A4 pq: Hp q 0 . = {}, Finally, the simple calculations show that A5 00 , i.e. the Ilyashenko attractor consists of a single point. Thus, = ÉÉÉ A1 A2 A3 A4 A5 , all inclusions being strict.

Exercise 3.5 Display graphically the attractors Aj of the system generated by equations (3.1) on the phase plane.

Exercise 3.6 Consider the dynamical system from Example 1.1 with ()= ()2 - = {}- ££ fx xx 1 . Prove that A1 x :1x 1 , = {}==; ± =={}= ± A3 x 0 x 1 , and A4 A5 x 1 . Ì Ì Exercise 3.7 Prove that A4 A3 and A5 A3 in general. Exercise 3.8 Show that all positive semitrajectories of a dynamical system which possesses a global minimal attractor are bounded sets.

In particular, the result of the last exercise shows that the global attractor can exist only under additional conditions concerning the behaviour of trajectories of the system at infinity. The main condition to be met is the dissipativity discussed in the next section. 24 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C h § 4 Dissipativity and Asymptotic a p Compactness t e r 1 From the physical point of view dissipative systems are primarily connected with ir- reversible processes. They represent a rather wide and important class of the dynamical systems that are intensively studied by modern natural sciences. These systems (unlike the conservative systems) are characterized by the existence of the accented direction of time as well as by the energy reallocation and dissipation. In particular, this means that limit regimes that are stationary in a certain sense can arise in the system when t ® + ¥ . Mathematically these features of the qualitative behaviour of the trajectories are connected with the existence of a bounded absorbing set in the phase space of the system. Ì (), A set B0 X is said to be absorbing for a dynamical system XSt if for = () ()Ì any bounded set BX in there exists t0 t0 B such that St B B0 for every ³ (), tt0 . A dynamical system XSt is said to be dissipative if it possesses a boun- (), ded absorbing set. In cases when the phase space XXS of a dissipative system t {}Î £ is a Banach space a ball of the form xX: x X R can be taken as an absorbing set. Therewith the value R is said to be a radius of dissipativity. As a rule, dissipativity of a dynamical system can be derived from the existence of a Lyapunov type function on the phase space. For example, we have the following assertion.

Theorem 4.1. (), Let the phase sspppaceace of a continuous dynamical system XSt be a Ba- nach space. Assume that: (a) there exists a continuous function Ux() on X possessing the properties j ()£ ()£ j() 1 x Ux 2 x , (4.1) j () j ()® ¥ where j r are continuous functions on R+ and 1 r + when r ® ¥ ; (b) there exist a derivative d US()y for t ³ 0 and positive numbers dt t ar and such that

d US()y £ - afor S y > r . (4.2) dt t t (), Then the dynamical system XSt is dissipative.

Proof. ³jr ()> ³ Let us choose R0 such that 1 r 0 for rR0 . Let = {}j () £ + l sup 2 r : r 1 R0 >j+ ()> > and R1 R0 1 be such that 1 r l for rR1 . Let us show that Dissipativity and Asymptotic Compactness 25

£ ³ £ St yR1 for allt 0 andyR0 . (4.3) Î £ Assume the contrary, i.e. assume that for some yX such that yR0 there > > exists a time t 0 possessing the property St yR1 . Then the continuity of Sty implies that there exists 0 < r ()£ ³ £ provided St y . It follows that USt y l for all tt0 . Hence, St yR1 for ³ all tt0 . This contradicts the assumption. Let us assume now that B is an arbitrary bounded set in XR that does not lie inside the ball with the radius 0 . Then equation (4.2) implies that ()£ ()- a £ - a Î USt y Uy tlB t , yB , (4.4) > r provided St y . Here = {}() Î lB sup Ux: xB. Î * < ()a- ¤ Let yB . If for a time t lB l the semitrajectory St y enters the ball with r £ ³ * the radius , then by (4.3) we have St yR1 for all tt . If that does not take place, from equation (4.4) it follows that l - l j ()£ ()£ ³ B 1 St y USt y l fort a , £ ³ a-1 ()- i.e.St yR1 fort lB l . Thus, l - l Ì {}£ ³ B St Bx: xR1 , t a .

This and (4.3) imply that the ball with the radius R1 is an absorbing set for the dy- (), namical system XSt . Thus, Theorem 4.1 is proved.

Exercise 4.1 Show that hypothesis (4.2) of Theorem 4.1 can be replaced by the requirement

d US()gy + US()y £ C , dt t t where g and C are positive constants.

Exercise 4.2 Show that the dynamical system generated in R by the diffe- \ rential equation x + fx()= 0 (see Example 1.1) is dissipative, provided the function fx() possesses the property: xf() x ³ dx2 - C , where d > 0 and CUx are constants (Hint: ()= x2 ). Find an upper estimate for the minimal radius of dissipativity.

Exercise 4.3 Consider a discrete dynamical system ()R, f n , where f is a continuous function on R . Show that the system ()R, f is dissipative, provided there exist r > 0 and 0 < r . 26 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C ()2, h Exercise 4.4 Consider a dynamical system R St generated (see Exam- a p ple 1.2) by the Duffing equation t \\ \ 3 e x ++e x x - ax = b , r where ab and are real numbers and e > 0 . Using the properties 1 of the function \ \ æö\ e Ux(), x = 1 x2 + 1 x4 - a x2 + n xx + x2 2 4 2 èø2 ()2, n> show that the dynamical system R St is dissipative for 0 small enough. Find an upper estimate for the minimal radius of dissipativity.

Exercise 4.5 Prove the dissipativity of the dynamical system generated by (1.4) (see Example 1.3), provided N N (),,,¼ £d-d 2 + > å xk fk x1 x2 xN å xk C , 0 . k = 1 k = 1

Exercise 4.6 Show that the dynamical system of Example 1.4 is dissipative if fz() is a bounded function.

Exercise 4.7 Consider a cylinder Ц with coordinates ()x, j , x Î R , j Î [01,) and the mapping T of this cylinder which is defined by the formula Tx(), j = ()x ¢j¢, , where x ¢a= xk+ sin2 pj, j¢= j + x ¢ ()mod1. Here a and k are positive parameters. Prove that the discrete dynamical system ()Ц , Tn is dissipative, provided 0 <

In the proof of the existence of global attractors of infinite-dimensional dissipative dynamical systems a great role is played by the property of asymptotic compactness. For the sake of simplicity let us assume that X is a closed subset of a Banach space. (), The dynamical system XSt is said to be asymptotically compact if for any > t 0 its evolutionary operator St can be expressed by the form = ()1 + ()2 St St St ,(4.5) ()1 ()2 where the mappings St and St possess the properties: Dissipativity and Asymptotic Compactness 27

a) for any bounded set BX in ()= ()1 ® ® ¥ rB t supSt y 0 , t + ; yBÎ X

b) for any bounded set BX in there exists t0 such that the set

[]g()2 ()= ()2 t B È St B (4.6) 0 ³ tt0 is compact in X , where []g is the closure of the set g . A dynamical system is said to be compact if it is asymptotically compact and ()1 º one can take St 0 in representation (4.5). It becomes clear that any finite-dimensional dissipative system is compact.

Exercise 4.9 Show that condition (4.6) is fulfilled if there exists a compact ()2 Ì set KH in such that for any bounded set B the inclusion St BK , ³ () tt0 B holds. In particular, a dissipative system is compact if it possesses a compact absorbing set.

Lemma 4.1. (), The dynamical system XSt is asymptotically compact if there exists a compact set K such that {}(), Î = limsup distSt uK : uB 0 (4.7) t ® ¥ for any set BX bounded in .

Proof. The distance to a compact set is reached on some element. Hence, for any > Î º ()2 Î t 0 and uX there exists an element vSt u K such that (), = - ()2 dist St uK St uSt u . ()1 = - ()2 Therefore, if we take St uSt uSt u , it is easy to see that in this case decomposition (4.5) satisfies all the requirements of the definition of asymptotic compactness.

Remark 4.1. In most applications Lemma 4.1 plays a major role in the proof of the property of asymptotic compactness. Moreover, in cases when the phase (), space XXS of the dynamical system t does not possess the structure of a linear space it is convenient to define the notion of the asymptotic (), compactness using equation (4.7). Namely, the system XSt is said to be asymptotically compact if there exists a compact K possessing property (4.7) for any bounded set BX in . For one more approach to the definition of this concept see Exercise 5.1 below. 28 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C h Exercise 4.10 Consider the infinite-dimensional dynamical system genera- a p ted by the retarded equation t \ e x ()t + axt()= fxt()()-1 , r where a > 0 and fz() is bounded (see Example 1.4). Show that 1 this system is compact.

Exercise 4.11 Consider the system of Lorentz equations arising as a three- mode Galerkin approximation in the problem of convection in a thin layer of liquid: \ ì x = -ss x + y , ï \ í y = rx- y - xz, ï \ î z = - bz+ xy. Here s , rb , and are positive numbers. Prove the dissipativity of the dynamical system generated by these equations in R3 . Hint: Consider the function Vx(),, y z = 1 ()x2 ++y2 ()zr-s- 2 2 on the trajectories of the system.

§ 5 Theorems on Existence of Global AAttttractortractor

For the sake of simplicity it is assumed in this section that the phase space X is a Banach space, although the main results are valid for a wider class of spaces (see, e. g., Exercise 5.8). The following assertion is the main result.

Theorem 5.1. (), Assume that a dynamical system XSt is dissipative and asymptoti- (), cally compact. Let BXS be a bounded absorbing set of the system t . Then the set A = w()B is a nonempty compact set and is a global attractor of the (), dynamical system XSt . The attractor AX is a connected set in .

In particular, this theorem is applicable to the dynamical systems from Exercises 4.2–4.11. It should also be noted that Theorem 5.1 along with Lemma 4.1 gives the following criterion: a dissipative dynamical system possesses a compact global attractor if and only if it is asymptotically compact. The proof of the theorem is based on the following lemma. Theorems on Existence of Global Attractor 29

Lemma 5.1. (), Let a dynamical system XSt be asymptotically compact. Then for any bounded set BX of the ww-limit set ()B is a nonempty compact invariant set.

Proof. () Let y ÎB . Then for any sequence {}t tending to infinity the set {S 2 y , n n tn n n =}12,,¼ is relatively compact, i.e. there exist a sequence n and an ele- () k ment yXÎ such that S 2 y tends to yk as ® ¥ . Hence, the asymptotic tn nk compactness gives us that k - £ ()1 + - ()2 ® ® ¥ ySt yn St yn ySt yn 0 ask . nk k nk k nk k =w() Thus, ySlim t yn . Due to Lemma 2.1 this indicates that B is non- k ® ¥ n k empty. k Let us prove the invariance of w -limit set. Let y Î w ()B . Then according {} ® ¥ {}Ì to Lemma 2.1 there exist sequences tn , tn , and zn B such that S z ® y . However, the mapping S is continuous. Therefore, tn n t S + z = S ° S z ® S y , n ® ¥ . ttn n t tn n t Î w() Lemma 2.1 implies that St y B . Thus, w()Ì w() > St B B , t 0 . Let us prove the reverse inclusion. Let y Î w()B . Then there exist sequences {}v Ì B and {}t : t ® ¥ such that S v ® y . Let us consider the se- n n n tn n quence y = S v , t ³ t . The asymptotic compactness implies that there n tn- t n n exist a subsequence t and an element zXÎ such that nk = ()2 zSlim t - t yn . k ® ¥ nk k As stated above, this gives us that = zSlim t - t yn . k ® ¥ nk k Therefore, z Î w()B . Moreover, === St zSlim t ° St - t vn lim St vn y . k ® ¥ nk k k ® ¥ nk k Îww() () Hence, ySt B . Thus, the invariance of the set B is proved. w() {} Let us prove the compactness of the set B . Assume that zn is a se- w() ³ quence in B . Then Lemma 2.1 implies that for any nt we can find n n and y Î B such that z - S y £ 1¤ n . As said above, the property of asymp- n n tn n {} totic compactness enables us to find an element zn and a sequence k such that

St y - z ® 0, k ® ¥ . nk nk 30 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C This implies that z Î w()B and z ®wz . This means that ()B is a closed and h nk a compact set in H . Lemma 5.1 is proved completely. p t e Now we establish Theorem 5.1. Let B be a bounded absorbing set of the dynamical r system. Let us prove that w()B is a global attractor. It is sufficient to verify that 1 w()B uniformly attracts the absorbing set B . Assume the contrary. Then the value {}(), w() Î ® ¥ sup dist St y B : yB does not tend to zero as t . This means that d > {}® ¥ there exist 0 and a sequence tn : tn such that ìü supíýdist ()S y, w()B : yBÎ ³ 2 d . îþtn Î Therefore, there exists an element yn B such that dist ()S y , w()B ³ d, n = 12,,¼ .(5.1) tn n As before, a convergent subsequence {}S y can be extracted from the sequence tn nk {}S y . Therewith Lemma 2.1 implies k tn n ºwÎ () zSlim t yn B k ® ¥ nk k which contradicts estimate (5.1). Thus, w()B is a global attractor. Its compactness follows from the easily verifiable relation º w()= ()2 A B Ç Ç St B . t > 0 t ³ t Let us prove the connectedness of the attractor by reductio ad absurdum. Assume

that the attractor AU is not a connected set. Then there exists a pair of open sets 1 and U2 such that ÇÆ¹ = , Ì È ÇÆ= Ui A , i 12 ,AU1 U2 ,U1 U2 . Let Ac = conv ()A be a convex hull of the set A , i.e.

ìüN N c = l Î l ³ l ==, ,,¼ A íýå i vi : vi A , i 0, å i 1 N 12 . îþ i = 1 i = 1 It is clear that Ac is a bounded connected set and Ac É A . The continuity of the c = Ì c mapping St implies that the set St A is also connected. Therewith ASt A St A . ÇÆc ¹ = , > Therefore, Ui St A , i 12 . Hence, for any t 0 the pair U1 , U2 cannot c = Î c cover St A . It follows that there exists a sequence of points xn Sn yn Sn A Ï È such that xn U1 U2 . The asymptotic compactness of the dynamical system enables us to extract a subsequence {}n such that x = S y tends in X to an k nk nk nk ® ¥ Ï È Î w()c element yk as . It is clear that yU1 U2 and y A . These equations w()c Ì w()= Ì È contradict one another since A B AU1 U2 . Therefore, Theorem 5.1 is proved completely. Theorems on Existence of Global Attractor 31

It should be noted that the connectedness of the global attractor can also be proved without using the linear structure of the phase space (do it yourself).

Exercise 5.1 Show that the assumption of asymptotic compactness in Theo- rem 5.1 can be replaced by the Ladyzhenskaya assumption: the sequence {}S u contains a convergent subsequence for any tn n {}Ì bounded sequence un X and for any increasing sequence {}Ì ® ¥ tn T+ such that tn + . Moreover, the Ladyzhenskaya assumption is equivalent to the condition of asymptotic compactness. (), Exercise 5.2 Assume that a dynamical system XSt possesses a compact global attractor AA . Let * be a minimal closed set with the property (), * = Î limdist St yA 0 for everyyX . t ® ¥ Then A* Ì A and A* = È{}w()x : xXÎ , i.e. A* coincides with the global minimal attractor (cf. Exercise 3.4).

Exercise 5.3 Assume that equation (4.7) holds. Prove that the global attractor A possesses the property A = w()K Ì K .

Exercise 5.4 Assume that a dissipative dynamical system possesses a global attractor A . Show that A = w ()B for any bounded absorbing set B of the system.

The fact that the global attractor A has the form A = w()B , where B is an absorbing set of the system, enables us to state that the set St B not only tends to the attractor At , but is also uniformly distributed over it as ® ¥ . Namely, the following assertion holds.

Theorem 5.2. (), Assume that a dissipative dynamical system XSt possesses a com- (), pact global attractor AB. Let be a bounded absorbing set for XSt . Then {}(), Î = limsup dist aSt B : aA 0 . (5.2) t ® ¥

Proof. {}Ì Assume that equation (5.2) does not hold. Then there exist sequences an Ì {}® ¥ A and tn : tn such that dist ()da , S B ³ for somed > 0 . (5.3) n tn {} The compactness of Aa enables us to suppose that n converges to an element aAÎ . Therewith (see Exercise 5.4) = , {}Ì aSlim t ym ym B , m ® ¥ m 32 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C {}t t® ¥ {} h where m is a sequence such that m . Let us choose a subsequence mn a such that tm ³ t + t for every n = 12,,¼ . Here t is chosen such that S B Ì p n n B B t Ì B for all tt³ . Let z = S y . Then it is clear that {}z Ì B and t B n tm - t m n e n n n r == aSlim t ym lim St zn . n ® ¥ mn n n ® ¥ n 1 Equation (5.3) implies that dist()a , S z ³³dist()a , S B d . n tn n n tn This contradicts the previous equation. Theorem 5.2 is proved.

For a description of convergence of the trajectories to the global attractor it is convenient to use the Hausdorff metric that is defined on subsets of the phase space by the formula r ()CD, = max{}hCD(), ; hDC(), ,(5.4) where CD, Î X and hCD(), = sup {}dist ()cD, : cCÎ .(5.5) Theorems 5.1 and 5.2 give us the following assertion.

Corollary 5.1. (), Let XSt be an asymptotically compact dissipative system. Then its global attractor A possesses the property limr ()S BA, = 0 for any t ® ¥ t (), bounded absorbing set BXS of the system t .

In particular, this corollary means that for any e > 0 there exists te > 0 such that > e for every tte the set St B gets into the -vicinity of the global attractor A ; e and vice versa, the attractor A lies in the -vicinity of the set St BB . Here is a bounded absorbing set. The following theorem shows that in some cases we can get rid of the requirement of asymptotic compactness if we use the notion of the global weak attractor.

Theorem 5.3. (), Let the phase space HHS of a dynamical system t be a separable (), Hilbert space. Assume that the system HSt is dissipative and its evolu- > tionary operator St is weakly closed, i.e. for all t 0 the weak convergence y ® y and S y ® z imply that zS= y . Then the dynamical system n t n t (), HSt possesses a global weak attractorattractor..

The proof of this theorem basically repeats the reasonings used in the proof of Theo- rem 5.1. The weak compactness of bounded sets in a separable Hilbert space plays the main role instead of the asymptotic compactness. Theorems on Existence of Global Attractor 33

Lemma 5.2. Assume that the hypotheses of Theorem 5.3 hold. For BHÌ we define ww() the weak -limit set w B by the formula w ()= () w B Ç È St B , (5.6) s ³ 0 ts³ w [] where Y w is the weak closure of the set Y . Then for any bounded set Ìw() BH the set w B is a nonempty weakly closed bounded invariant set.

Proof. The dissipativity implies that each of the sets gs ()B = []S ()B is w Èts³ t w bounded and therefore weakly compact. Then the Cantor theorem on the collection of nested compact sets gives us that w ()B = gs ()B is a non- w Çs ³ 0 w Î w () empty weakly closed bounded set. Let us prove its invariance. Let y w B . Then there exists a sequence y Î S ()B such that y ® y weakly. The n Ètn³ t n {} dissipativity property implies that the set St yn is bounded when t is large {} enough. Therefore, there exist a subsequence yn and an element z such ® ® k that yn y and St yn z weakly. The weak closedness of St implies that = k Îkgs () ³ Î gs () zSt y . Since St yn w B for nk s , we have that z w B for all s . Î w () k w ()Ì w() Hence, z w B . Therefore, St w B w B . The proof of the reverse inclusion is left to the reader as an exercise.

For the proof of Theorem 5.3 it is sufficient to show that the set = w () Aw w B , (5.7) (), where BHS is a bounded absorbing set of the system t , is a global weak attractor for the system. To do that it is sufficient to verify that the set B is uniformly attract- = w () ed to Aw w B in the weak topology of the space H . Assume the contrary. Then O {}Ì { ® there exist a weak vicinity of the set Aw and sequences yn B and tn : tn ®}¥ such that S y Ï O . However, the set {}S y is weakly compact. There- tn n tn n Ï O {} fore, there exist an element z and a sequence nk such that = - zwlim St yn . k ® ¥ nk k s s However, St y Î g ()B for t ³ s . Thus, z Î g ()B for all s ³ 0 and z Î nk nk w nk w Î w () w B , which is impossible. Theorem 5.3 is proved.

Exercise 5.5 Assume that the hypotheses of Theorem 5.3 hold. Show that

the global weak attractor Aw is a connected set in the weak topology of the phase space H . * = {w () Exercise 5.6 Show that the global weak minimal attractor Aw È w x : xHÎ} is a strictly invariant set. 34 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C h Exercise 5.7 Prove the existence and describe the structure of global and a p global minimal attractors for the dynamical system generated by t the equations e r \ ì x = m xy- - xx()2 + y2 , í \ 1 î y = x + m yyx- ()2 + y2 for every real m . (), Exercise 5.8 Assume that XXS is a metric space and t is an asymptotically compact (in the sense of the definition given in Remark 4.1) dynamical system. Assume also that the attracting compact K is contained in some bounded connected set. Prove the validity of the assertions of Theorem 5.1 in this case.

In conclusion to this section, we give one more assertion on the existence of the global attractor in the form of exercises. This assertion uses the notion of the asymptotic (), smoothness (see [3] and [9]). The dynamical system XSt is said to be asympto- ()Ì ³ tically smooth if for any bounded positively invariant St BB, t 0 set Ì (), ® ® ¥ BX there exists a compact KhS such that t BK 0 as t , where the value h ().., is defined by formula (5.5).

Exercise 5.9 Prove that every asymptotically compact system is asymptotically smooth. (), Exercise 5.10 Let XSt be an asymptotically smooth dynamical system. Assume that for any bounded set BXÌg the set +B = = () (), Èt ³ 0 St B is bounded. Show that the system XSt possesses a global attractor A of the form AwB= È{}(): BXÌ , B is bounded .

Exercise 5.11 In addition to the assumptions of Exercise 5.10 assume that (), XSt is pointwise dissipative, i.e. there exists a bounded set Ì (), ® ® ¥ B0 X such that dist X St yB0 0 as t for every point yXÎ . Prove that the global attractor A is compact.

§ 6 On the Structure of Global Attractor

The study of the structure of global attractor of a dynamical system is an important problem from the point of view of applications. There are no universal approaches to this problem. Even in finite-dimensional cases the attractor can be of complicated structure. However, some sets that undoubtedly belong to the attractor can be poin- On the Structure of Global Attractor 35

ted out. It should be first noted that every stationary point of the semigroup St belongs to the attractor of the system. We also have the following assertion.

Lemma 6.1. Assume that an element zA lies in the global attractor of a dynamical (), g system XSt . Then the point z belongs to some trajectory that lies in A wholly.

Proof. = Î {}Ì Since St AA and zA , then there exists a sequence zn A such = = = ,,¼ that z0 z ,S1zn zn -1 ,n 12 . Therewith for discrete time the re- g = {}Î = ³ = quired trajectory is un : n Z , where un Sn z for n 0 and un = £ z-n for n 0 . For continuous time let us consider the value , ³ ì St z t 0, ut()= í , - ££- + , = ,,¼ î Stn+ zn nt n1 n 12

Then it is clear that ut()Î A for all t Î R and St ut()=tut()+ t for ³ 0 , t Î R . Therewith u ()0 = z . Thus, the required trajectory is also built in the continuous case.

Exercise 6.1 Show that an element z belongs to a global attractor if and only if there exists a bounded trajectory g = {}ut(): -¥ <

Unstable sets also belong to the global attractor. Let Y be a subset of the phase (), space XXS of the dynamical system t . Then the unstable set emanating () Î from Y is defined as the set M+ Y of points zX for every of which there exists a trajectory g = {}ut(): t Î T such that u ()0 ==zut, lim dist ()(), Y 0 . t ® -¥

() ()= () Exercise 6.2 Prove that M+ Y is invariant, i.e. St M+ Y M+ Y for all t > 0 .

Lemma 6.2. N (), Let be a set of stationary points of the dynamical system XSt ()N Ì possessing a global attractor A . Then M+ A .

Proof. N = {}= , > It is obvious that the set z : St zz t 0 lies in the attractor Î ()N of the system and thus it is bounded. Let z M+ . Then there exists a tra- g = {}(), Î ()= jectory z ut t T such that u 0 z and dist()u()t , N ® 0, t¥® - . 36 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C = {}()t t £ - > h Therefore, the set Bs u : s is bounded when s 0 is large a ® ¥ enough. Hence, the set St Bs tends to the attractor of the system as t + . p Î ³ t However, zSt Bs for ts . Therefore, e r (), £ {}(), Î ® , ® ¥ dist zA sup dist St yA: yBs 0 t + . 1 This implies that zAÎ . The lemma is proved.

Exercise 6.3 Assume that the set N of stationary points is finite. Show that l ()N = () M+ È M+ zk , k = 1 () where zk are the stationary points of St (the set M+ zk is called an unstable manifold emanating from the stationary point zk ). ()N Thus, the global attractor A includes the unstable set M+ . It turns out that under certain conditions the attractor includes nothing else. We give the following defi- Ì > nition. Let YS be a positively invariant set of a semigroup t : StYY , t 0 . The continuous functional D()y defined on Y is called the Lyapunov function of the (), dynamical system XSt on Y if the following conditions hold: ÎD() a) for any yY the function St y is a nonincreasing function with respect to t ³ 0 ; > ÎD()= D() b) if for some t0 0 and yX the equation y St y holds, then = ³ 0 ySt y for all t 0 , i.e. yS is a stationary point of the semigroup t .

Theorem 6.1. (), Let a dynamical system XSt possess a compact attractor A. Assume D() = ()N also that the Lyapunov function y exists on AA. Then M+ , where N is the set of stationary points of the dynamical system.

Proof. Let yAÎg . Let us consider a trajectory passing through y (its existence follows from Lemma 6.1). Let – g =g{}ut(): t Î T andt = {}ut(): t £ t . – – Since gt ÌgA , the closure []t is a compact set in X . This implies that the a -limit set ag()= [] g– Ç t t < 0 of the trajectory gag is nonempty. It is easy to verify that the set () is invariant: ag()=D ag() () ag() St . Let us show that the Lyapunov function y is constant on . Î ag() {}- ¥ Indeed, if u , then there exists a sequence tn tending to such that On the Structure of Global Attractor 37

lim ut()= u . ® -¥ n tn Consequently, D()u = lim D()ut(). n ® ¥ n By virtue of monotonicity of the function D()u along the trajectory we have D()u = sup {} D()u ()t : t < 0 . Therefore, the function D()u is constant on ag() . Hence, the invariance of the set ag() D()D= () > Îagag() () gives us that St u u , t 0 for all u . This means that lies in the set N of stationary points. Therewith (verify it yourself) limdist()ut(), ag() = 0 . t ® -¥ Î ()N Hence, y M+ . Theorem 6.1 is proved.

Exercise 6.4 Assume that the hypotheses of Theorem 6.1 hold. Then for any element yAÎw its -limit set w()y consists of stationary points of the system.

Thus, the global attractor coincides with the set of all full trajectories connecting the stationary points.

(), Exercise 6.5 Assume that the system XSt possesses a compact global attractor and there exists a Lyapunov function on X . Assume that the Lyapunov function is bounded below. Show that any semitrajectory of the system tends to the set N of stationary points of the system as t ® + ¥ , i.e. the global minimal attractor coincides with the set N .

In particular, this exercise confirms the fact realized by many investigators that the global attractor is a too wide object for description of actually observed limit regimes of a dynamical system.

Exercise 6.6 Assume that ()R, S is a dynamical system generated by the t \ logistic equation (see Example 1.1): x + a xx()-1 = 0, a > 0 . Show that Vx()= x3 ¤ 3 - x2 ¤ 2 is a Lyapunov function for this system.

Exercise 6.7 Show that the total energy \ \ Ex(), x = 1 x2 + 1 x4 - a x2 - bx 2 4 2 is a Lyapunov function for the dynamical system generated (see Example 1.2) by the Duffing equation \\ \ x ++e x x3 - ax =eb , > 0 . 38 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C h If in the definition of a Lyapunov functional we omit the second requirement, then a a minor modification of the proof of Theorem 6.1 enables us to get the following as- p t sertion. e r Theorem 6.2. 1 (), Assume that a dynamical system XSt possesses a compact global attractor A and there exists a continuous function Y()y on X such that Y() Î L St y does not increase with respect to tyX for any . Let be a set of elements uAÎY such that ()Yut() =¥()u for all - <

Proof. ()L = In fact, the property M+ A was established in the proof of Theorem 6.1. As to the property A* ÌYL , it follows from the constancy of the function ()u on the ww -limit set ()x of any element xXÎ .

Exercise 6.8 Apply Theorem 6.2 to justify the results of Example 3.1 (see also Exercise 4.8).

N (), If the set of stationary points of a dynamical system XSt is finite, then Theo- rem 6.1 can be extended a little. This extension is described below in Exercises 6.9– (), 6.12. In these exercises it is assumed that the dynamical system XSt is continuous and possesses the following properties: (a) there exists a compact global attractor A ; (b) there exists a Lyapunov function D()x on A ; N =D{}, ¼ ()¹ (c) the set z1 zN of stationary points is finite, therewith zi ¹ D() ¹ zj for ij and the indexing of zj possesses the property D()<<< D() ¼ D() z1 z2 zN .(6.1) We denote j = () = ,,¼ , = Æ Aj È M+ zk , j 12 N ,A0 . k = 1

= = ,,¼ Exercise 6.9 Show that St Aj Aj for all j 12 N . Ì {} Exercise 6.10 Assume that BAj\ zj . Then {}(), Î = limsup dist St yAj -1 : yB 0 .(6.2) t ® ¥

Exercise 6.11 Assume that the function D is defined on the whole X . Then Ì {}D()< D()- d (6.2) holds for any bounded set Bx: x zj , where d is a positive number. On the Structure of Global Attractor 39

[]() () Exercise 6.12 Assume that M+ zj is the closure of the set M+ zj and ¶ ()= []() () ¶Ì M+ zj M+ zj \ M+ zj is its boundary. Show that M+zj Ì Aj -1 and []()= []() ¶ ()= ¶ () St M+ zj M+ zj ,.St M+ zj M+ zj

It can also be shown (see the book by A. V. Babin and M. I. Vishik [1]) that under some additional conditions on the evolutionary operator St the unstable manifolds () 1 M+ zj are surfaces of the class C , therewith the facts given in Exercises 6.9–6.12 remain true if strict inequalities are substituted by nonstrict ones in (6.1). It should be noted that a global attractor possessing the properties mentioned above is frequently called regular. Let us give without proof one more result on the attractor of a system with a finite number of stationary points and a Lyapunov function. This result is important for applications. At first let us remind several definitions. Let S be an operator acting in a Ba- nach space XS . The operator is called Frechét differentiable at a point xXÎ provided that there exists a linear bounded operator S ¢()x : XX® such that Sy()- Sx()-gS¢()x ()yx- £ ()xy- xy- for all y from some vicinity of the point x, where gx()®x0 as ® 0 . Therewith, the operator SC is said to belong to the class 1+ a, 0 < 1 is finite-dimensional.

Theorem 6.3. Let X be a Banach space and let a continuous dynamical system (), XSt possess the properties: 1) there exists a global attractor A ; 2) there exists a vicinity W of the attractor A such that - £ a ()t - t - St xSt y Ce St xSt y ³³t W ³ for all t 0 , provided St xS and t y belong to for all t 0 ; 3) there exists a Lyapunov function continuous on X ; N = {},,¼ 4) the set z1 zN of stationary points is finite and all the points are hyperbolic; 40 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C (), ® h 5) the mapping tu St u is continuous. a Then for any compact set BX in the estimate p t ìü e sup íýdist ()S yA, : yBÎ £ C e-h t (6.3) r îþt B 1 holds for all t ³h0 , where > 0 does not depend on B .

The proof of this theorem as well as other interesting results on the asymptotic behaviour of a dynamical system possessing a Lyapunov function can be found in the book by A. V. Babin and M. I. Vishik [1]. To conclude this section, we consider a finite-dimensional example that shows how the Lyapunov function method can be used to prove the existence of periodic trajectories in the attractor.

Example 6.1 (on the theme by E. Hopf) Studying Galerkin approximations in a model suggested by E. Hopf for the description of possible mechanisms of turbulence appearence, we obtain the following system of ordinary differential equations \ ì u +++m uv2 w2 = 0, (6.4) ï \ í v + n vvu-b- w = 0, (6.5) ï \ î w ++n wwu-bv = 0. (6.6) Here mnb is a positive parameter, and are real parameters. It is clear that the Cauchy problem for (6.4)–(6.6) is solvable, at least locally for any initial condition. Let us show that the dynamical system generated by equations (6.4)–(6.6) is dissipative. It will also be sufficient for the proof of global solvability. Let us introduce a new unknown function u* = u + m¤n2 - . Then equations (6.4)– (6.6) can be rewritten in the form

ì \ æöm ï u* +++mm u* v2 w2 = - n , ï èø2 ï \ í v + 1 m vvu-b* - w = 0, ï 2 ï \ ï w ++1 m wwu-b* v = 0. î 2

These equations imply that

2 2 2 2 m 2 2 æöm 1 d ()mu* ++v w ++u* ()mv + w = - n u* 2 dt 2 èø2 on any interval of existence of solutions. Hence, 2 æö2 2 2 2 2 2 æöm d ()u* ++v w +mm ()u* ++v w £ - n . dt èø èø2 On the Structure of Global Attractor 41

Thus, u*()t 2 ++vt()2 wt()2 £ 2 æö2 2 2 æöm -m £ u*()0 ++v()0 w()0 e-m t + - n ()1 -e t . èøèø2

Firstly, this equation enables us to prove the global solvability of problem (6.4)– (6.6) for any initial condition and, secondly, it means that the set ìüm 2 m 2 = (),, æö+ - n ++2 2 £ + æö - n B0 íýuvw: èøu v w 1 èø îþ2 2 ()3, is absorbing for the dynamical system R St generated by the Cauchy problem for equations (6.4)–(6.6). Thus, Theorem 5.1 guarantees the existence of a global attractor A . It is a connected compact set in R3 .

()3, Exercise 6.13 Verify that B0 is a positively invariant set for R St .

In order to describe the structure of the global attractor A we introduce the polar coordinates vt()= rt()cos j()t ,wt()= rt()sin j()t on the plane of the variables {}vw; . As a result, equations (6.4)–(6.6) are trans- formed into the system \ ì u ++m ur2 = 0, (6.7) í \ î r + n rur- = 0, (6.8) j()= -j b + { = , therewith, t t 0 . System (6.7) and (6.8) has a stationary point u 0 r =}0 for all m>n0 and ÎnR . If < 0 , then one more stationary point {u = n, r =}-mn occurs in system (6.7) and (6.8). It corresponds to a periodic trajectory of the original problem (6.4)–(6.6).

Exercise 6.14 Show that the point ()00; is a stable node of system (6.7) and (6.8) when n >n0 and it is a saddle when < 0 .

Exercise 6.15 Show that the stationary point {}u ==n,mnr - is stable m (n 0 , then (6.7) and (6.8) imply that

1 d u2 + r2 + min ()mn, ()u2 + r2 £ 0 . 2 dt Therefore, - ()mn, ut()2 + rt()2 £ u()0 2 + r()0 2 e 2min t . 42 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C n > ()3, h Hence, for 0 the global attractor A of the system R St consists of the single a stationary exponentially attracting point p t {}u ==0, v 0, w =0 . e r Exercise 6.16 Prove that for n = 0 the global attractor of problem (6.4)– 1 (6.6) consists of the single stationary point {}u = 0, v ==0, w 0 . Show that it is not exponentially attracting.

Now we consider the case n < 0 . Let us again refer to problem (6.7) and (6.8). It is clear that the line r = 0 is a stable manifold of the stationary point {}u ==0, r 0 . ()> () Moreover, it is obvious that if rt0 0 , then the value rt remains positive for all > tt0 . Therefore, the function Vu(), r = 1 ()u - n 2 ++1 r2 mn ln r (6.9) 2 2 is defined on all the trajectories, the initial point of which does not lie on the line {}r = 0 . Simple calculations show that

d ()mVut()(), rt() + ()ut()- n2 = 0 (6.10) dt and æö2 2 Vu(), r ³ V()nmn, - + 1 u - n + r - -mn ;(6.11) 2 èø therewith, V()nmn, - = ()mn12¤ ln()e¤ ()mn . Equation (6.10) implies that the function Vu(), r does not increase along the trajectories. Therefore, any semi- {}()(); (), Î {}, ; ¹ trajectory ut rt t R+ emanating from the point u0 r0 r0 0 of ()´ , the system RR+ St generated by equations (6.7) and (6.8) possesses the ()(), () £ (), ³ property Vut rt Vu0 r0 for t 0 . Therewith, equation (6.9) implies that this semitrajectory can not approach the line {}r = 0 at a distance less then {}[]¤ ()mn × (), = { = n, exp 1 Vu0 r0 . Hence, this semitrajectory tends to yu r =}-mn . Moreover, for any x Î R the set

Bx = {}yur= (), : Vu()x, r £ e > = ()xe, is uniformly attracted to y , i.e. for any 0 there exists t0 t0 such that Ì {}-e£ St Bx y : yy . e > ® ¥ Î Indeed, if it is not true, then there exist 0 0 , a sequence tn + , and zn Bx such that S z -ey > . The monotonicity of Vy() and property (6.11) imply that tn n 0 1 2 VS()z ³³VS()z V()nmn, - + e t n tn n 2 0 ££ {} for all 0 ttn . Let zz be a limit point of the sequence n . Then after passing to the limit we find out that On the Structure of Global Attractor 43

Fig. 2. Qualitative behaviour of solutions to problem (6.7), (6.8): a) -m¤ 8 <

1 2 VS()z ³ V()nmn, - + e , t ³ 0 t 2 0 Ï {}= ® = { =,n with zr0 . Thus, the last inequality is impossible since St zy u r =}-mn . Hence {}(), Î = limsup dist St yy: yBx 0 .(6.12) t ® ¥ The qualitative behaviour of solutions to problem (6.7) and (6.8) on the semiplane is shown on Fig. 2. In particular, the observations above mean that the global minimal attractor ()3, Amin of the dynamical system R St generated by equations (6.4)–(6.6) consists of the saddle point {}u ==0, v 0, w =0 and the stable limit cycle 2 2 Cn = {}u ==n,mn v + w - (6.13) for n < 0 . Therewith, equation (6.12) implies that the cycle Cn uniformly attracts all bounded sets B in R3 possessing the property dvº inf{}2 + w2 : ()uvw,, Î B > 0 ,(6.14) i.e. which lie at a positive distance from the line {}v ==0, w 0 .

Exercise 6.17 Using the structure of equations (6.7) and (6.8) near the stationary point {}u = n, r = -mn , prove that a bounded set B possessing property (6.14) is uniformly and exponentially attracted to the cycle Cn , i.e. -g ()tt- {}(), , Î £ B sup distSt yCn yB Ce ³g for ttB , where is a positive constant. 44 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C = (),, ()3, h Now let y0 u0 v0 w0 lie in the global attractor A of the system R St . a ¹ 2 =mn2 + 2 ¹ - Assume that r0 0 and r0 v0 w0 . Then (see Lemma 6.1) there exists p g = {}()= ()(); (); (), Î ()= t a trajectory yt ut vt wt t R lying in Ay such that 0 y0 . e The analysis given above shows that yt()® C as t ® + ¥ . Let us show that r n yt()® 0 when t ® -¥ . Indeed, the function Vut()(), rt() is monotonely nonde- 1 creasing as t ® -¥ . If we argue by contradiction and use the fact that yt() is bounded we can easily find out that lim Vut()(), rt() = ¥ t ® -¥ and therefore 12 ()= æö()2 + ()2 ® ® -¥ rt èøvt wt 0 as t . (6.16) Equation (6.7) gives us that t -m ()- t -m ()- t ut()= e t us()- ò e t []r()t 2 dt .(6.17) s Since us() is bounded for all s Î R , we can get the equation t -m ()- t ut()= - ò e t []r ()t 2 dt -¥ by tending s ® – ¥ in (6.17). Therefore, by virtue of (6.16) we find that ut()® 0 as t ® – ¥ . Thus, yt()® 0 as t ®n– ¥ . Hence, for < 0 the global attractor A

Fig. 3. Attractor of the system (6.4)–(6.6); a) -nm¤ 8 <<0 , b) nm< - ¤ 8 Stability Properties of Attractor and Reduction Principle 45

()3, () of the system R St coincides with the union of the unstable manifold M+ 0 emanating from the point {}u ==0, v 0, w =0 and the limit cycle (6.13). The attractor is shown on Fig. 3.

§ 7 Stability Properties of Attractor and Reduction Principle

(), A positively invariant set MXS in the phase space of a dynamical system t is said to be stable (in Lyapunov’s sense) in X if its every vicinity O contains some O ¢ ()O ¢ Ì O ³ vicinity such that St for all t 0 . Therewith, M is said to be asymp- ® ® ¥ Î O ¢ totically stable if it is stable and St yM as t for every y . A set M is called uniformly asymptotically stable if it is stable and {}(), Î O ¢ = limsup distSt yM : y 0 . (7.1) t ® ¥ The following simple assertion takes place.

Theorem 7.1. Let A be the compact global attractor of a continuous dynamical sys- (), tem XSt . Assume that there exists its bounded vicinity U such that the (), ® ´ mapping tu St u is continuous on R+ U. Then A is a stable set.

Proof. O > Ì O Assume that is a vicinity of AT . Then there exists 0 such that St U for tT³ . Let us show that there exists a vicinity O ¢ of the attractor A such that O ¢ Ì O Î [], {} St for all t 0 T . Assume the contrary. Then there exist sequences un and {}t such that dist()u , A ® 0 , {}t Î[]0, T and S u ÏO . The set A being n n n tn n compact, we can choose a subsequence {}n such that u ® u Î A as t ® k nk nk ® Î [], (), ® t 0 T . Therefore, the continuity property of the function tu St u gives us that St u ® S u Î A . This contradicts the equation S u ÏO . Thus, nk nk t tn n O ¢ O ¢ Ì O Î [], there exists such that St for t 0 T . We can choose T such that ()O ¢ Ç Ì O ³ St U for all t 0 . Therefore, the attractor A is stable. Theorem 7.1 is proved.

It is clear that the stability of the global attractor implies its uniform asymptotic stability.

Exercise 7.1 Assume that M is a positively invariant set of a system (), Ï XSt . Prove that if there exists an element yM such that its aa-limit set ()y possesses the property a()y ÇÆM ¹ , then M is not stable. 46 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C h In particular, the result of this exercise shows that the global minimal attractor can a appear to be an unstable set. p t e Exercise 7.2 Let us return to Example 3.1 (see also Exercises 4.8 and 6.8). r Show that: 1 (a) the global attractor A1 and the Milnor attractor A4 are stable;

(b) the global minimal attractor A3 and the Ilyashenko attractor A5 are unstable.

Now let us consider the question concerning the stability of the attractor with respect to perturbations of a dynamical system. Assume that we have a family of dy- (), l namical systems XSt with the same phase space X and with an evolutionary l l operator St depending on a parameter which varies in a complete metric space L . The following assertion was proved by L. V. Kapitansky and I. N. Kostin [6].

Theorem 7.2. (), l Assume that a dynamical system XSt possesses a compact global attractor Al for every lLÎ . Assume that the following conditions hold: (a) there exists a compact KXÌ such that Al ÌlLK for all Î ; l l (b) if l ® l , x Î A k and x ® x , then S k x ® S x for some k 0 k k 0 t k t0 0 > 0 t0 0 . l l Then the family of aattttttractorsractors A is upper semicontinuous at the point 0 , i.e. l l l l hA()k, A 0 º sup {}dist ()yA, 0 : yAÎ k ® 0 (7.2) l ® l as k 0 .

Proof. Assume that equation (7.2) does not hold. Then there exist a sequence l ® l l k ® l Î k (), 0 ³dd > 0 and a sequence xk A such that dist xk A for some 0 . But the sequence x lies in the compact K . Therefore, without loss of generality we can k l ® Î Î Ï 0 assume that xk x0 K for some x0 K and x0 A . Let us show that this result leads to contradiction. Let g = {}u ()t :–¥ < where u0 x0 . Here t0 0 is a fixed number. Sequential application of condition (b) gives us that Stability Properties of Attractor and Reduction Principle 47

l l ==()-()- kn() ()- =0 uml- lim ukn() mlt0 lim Slt ukn() mt0 Slt um n ® ¥ n ® ¥ 0 0 for all m = 12,,¼ and l = 12,,¼ ,m . It follows that the function

l ì 0 , ³ ï St u0 t 0, ()= ut í l ï 0 S + u , -t m £ t < -t ()m - 1 , m = 12,,¼ î tt0 m m 0 0 gives a full trajectory g passing through the point x . It is obvious that the trajectory 0 l g is bounded. Therefore (see Exercise 6.1), it wholly belongs to A 0 , but that con- Ï l tradicts the equation x0 A 0 . Theorem 7.2 is proved. ll® Exercise 7.3 Following L. V. Kapitansky and I. N. Kostin [6], for 0 ()l L l L define the upper limit A 0 ; of the attractors A along by the equality

()l , L = {}l lLÎ , <<()ll, d A 0 Ç È A : 0 dist 0 , d > 0 where []. denotes the closure operation. Prove that if the hypotheses of Theorem 7.2 hold, then A ()l , L is a nonempty compact in- l 0 variant set lying in the attractor A 0 .

Theorem 7.2 embraces only the upper semicontinuity of the family of attractors {}Al . In order to prove their continuity (in the Hausdorff metric defined by equation (5.4)), additional conditions should be imposed on the family of dynamical systems (), l XSt . For example, the following assertion proved by A. V. Babin and M. I. Vishik l l concerning the power estimate of the deviation of the attractors A and A 0 in the Hausdorff metric holds.

Theorem 7.3. (), l Assume that a dynamical system XSt possesses a global attractor Al for every lLÎ . Let the following conditions hold: Ì l ÌlLÎ (a) there exists a bounded set B0 X such that A B0 for all and ()l , l £L-h t,lÎ hSt B0 A C0 e , (7.3) >h>l with constants C0 0 and 0 independent of and with hB(), A= sup {}dist ()bA, : bBÎ ; l ,Ll Î , Î (b) for any 1 2 and u1 u2 B0 the estimate l l ()1 , 2 £ a t ()(), + ()l , l dist St u1 St u2 C1e dist u1 u2 dist 1 2 (7.4) 48 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C al h holds, with constants C1 and independent of . a > p Then there exists C2 0 such that t l l q h e r ()A 1, A 2 £ C []dist()l , l , q = . (7.5) r 2 1 2 ha+ 1 Here r ().., is the Hausdorff metric defined by the formula r ()BA, = max {}hB(), A; hAB(), .

Proof. By virtue of the symmetry of (7.5) it is sufficient to find out that l l q ()1, 2 £ []()l , l hA A C2 dist 1 2 .(7.6) Equation (7.3) implies that for any e > 0 l ÌlLO ()l Î St B0 e A for all (7.7) ³ *()he, º -1 ()¤ e + O ()l e when tt C0 ln1 lnC0 . Here e A is an -vicinity of the set Al . It follows from equation (7.4) that l l l l hS()1 B , S 2 B = sup inf dist ()S 1 xS, 2 y £ t 0 t 0 Î Î t t xB0 yB0 l l ££sup dist()S 1 xS, 2 x C ea t dist()l , l . (7.8) Î t t 1 1 2 xB0 l Ì l = l l Ì l ³ *()e, Since A B0 , we have A St A St B0 . Therefore, with tt C0 , equation (7.7) gives us that l ÌÌl O ()l A St B0 e A .(7.9) For any xz, Î X the estimate dist()xA, l £ dist()xz, + dist()zA, l holds. Hence, we can find that dist()xA, l £ dist()xz, + e l for all xXÎ and z Î O e()A . Consequently, equation (7.9) implies that (), l £ (), l + e, Î dist xA dist xSt B0 x X ³ *()e, for tt C0 . It means that l l l hA()1, A 2 = supdist ()xA, 2 £ l xAÎ 1 l l l £ sup dist ()xS, 2 B + e £ hS()e1 B , S 2 B + . l t 0 t 0 t 0 xAÎ 1 Thus, equation (7.8) gives us that for any e > 0 Stability Properties of Attractor and Reduction Principle 49

l l hA()1, A 2 £ C ea t dist ()l , l + e 1 1 2 h ³el*()e, = [](), l q = = *()ºe, for tt C0 . By taking dist 1 2 , q ha+ and tt C0 º h-1 ()¤ e + ln1 lnC0 in this formula we find estimate (7.6). Theorem 7.3 is proved.

It should be noted that condition (7.3) in Theorem 7.3 is quite strong. It can be veri- fied only for a definite class of systems possessing the Lyapunov function (see Theo- rem 6.3). In the theory of dynamical systems an important role is also played by the notion of the Poisson stability. A trajectory g = {}ut(): -¥ <

Exercise 7.4 Show that any Poisson stable trajectory is contained in the global minimal attractor if the latter exists.

Exercise 7.5 A trajectory g is Poisson stable if and only if any point x of this trajectory is recurrent, i.e. for any vicinity O ' x there exists > Î O t 0 such that St x .

The following exercise testifies to the fact that not only periodic (and stationary) trajectories can be Poisson stable.

() Exercise 7.6 Let Cb R be a Banach space of continuous functions boun- ()(), ded on the real axis. Let us consider a dynamical system Cb R St with the evolutionary operator defined by the formula ()()=,()+ ()Î () St f x fx t fx Cb R . ()= w + w Show that the element f0 x sin 1 x sin 2 x is recurrent for w w w ¤ w any real 1 and 2 (in particular, when 1 2 is an irrational g = {}()+ -¥ <<¥ number). Therewith the trajectory f0 xt: t is Poisson stable.

In conclusion to this section we consider a theorem that is traditionally associated with the stability theory. Sometimes this theorem enables us to significantly decrease the dimension of the phase space, this fact being very important for the study of infinite-dimensional systems.

Theorem 7.4. (reduction principle). (), Assume that in a dissipative dynamical system XSt there exists a positively invariant locally compact set M possessing the property of uniform attraction, i.e. for any bounded set BXÌ the equation 50 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C (), = h limsup dist St yM 0 (7.10) a t ® ¥ yBÎ p (), t holds. Let A be a global attractor of the dynamical system MSt . Then A e is also a global attractor of ()XS, . r is also a global attractor of t .

1 Proof. It is sufficient to verify that (), = limsup dist St yA 0 (7.11) t ® ¥ yBÎ for any bounded set BXÌ . Assume that there exists a set B such that (7.11) does {}Ì {}® ¥ not hold. Then there exist sequences yn B and tn : tn such that dist()S y , A ³ d (7.12) tn n d > (), for some 0 . Let B0 be a bounded absorbing set of XSt . We choose a moment t0 such that ìüd (), Î Ç £ sup íýdist St yA: yMB0 .(7.13) îþ0 2 (), This choice is possible because AMS is a global attractor of t . Equation (7.10) implies that

dist()S - y , M ® 0, t ® ¥ . tn t0 n n

The dissipativity property of ()XS, gives us that S - y Î B when n is large t tn t0 n 0 enough. Therefore, local compactness of the set M guarantees the existence of an Î Ç {} element zMB0 and a subsequence nk such that = zSlim t - t yn . k ® ¥ nk 0 k

This implies that St y ® St z . Therefore, equation (7.12) gives us that nk nk 0 dist()S zA, ³ d . By virtue of the fact that zMBÎ Ç this contradicts equation t0 0 (7.13). Theorem 7.4 is proved.

Example 7.1 We consider a system of ordinary differential equations ì \ y + y3 - y = yz2, y = y , ï t = 0 0 í \ (7.14) ï z + z ()1+ y2 ==0, z z . î t = 0 0 (), It is obvious that for any initial condition y0 z0 problem (7.14) is uniquely (), *(), solvable over some interval 0 t y0 z0 . If we multiply the first equation by yz and the second equation by and if we sum the results obtained, then we get that Stability Properties of Attractor and Reduction Principle 51

1 d ()y2 + z2 + y4 - y2 + z2 = 0 , t Î ()0, t*()y , z . 2 dt 0 0 This implies that the function Vy(), z = y2 + z2 possesses the property

d Vyt()(), zt() + 2 Vyt()(), zt() £ 2 , t Î [0, t*())y , z . dt 0 0 Therefore, ()(), () £ (), -2 t + Î [ , *()), Vyt zt Vy0 z0 e 1 , t 0 t y0 z0 . This implies that any solution to problem (7.14) can be extended to the whole ()2, semiaxis R+ and the dynamical system R St generated by equation (7.14) is dissipative. Obviously, the set My= {}(), 0 : y Î R is positively invariant. Therewith the second equation in (7.14) implies that

1 d z2 + z2 £ 0 ,.t > 0 2 dt ()2 £ 2 -2 t Hence, zt z0 e . Thus, the set M exponentially attracts all the bounded sets in R2 . Consequently, Theorem 7.4 gives us that the global attractor of (), ()2, the dynamical system MSt is also the attractor of the system R St . But on the set M system of equations (7.14) is reduced to the differential equation \ y + y3 - y = 0, y = y .(7.15) t = 0 0 Thus, the global attractors of the dynamical systems generated by equations (7.14) and (7.15) coincide. Therewith the study of dynamics on the plane is reduced to the investigation of the properties of the one-dimensional dynamical system.

Exercise 7.7 Show that the global attractor A of the dynamical system ()2, R St generated by equations (7.14) has the form Ayz= {}(), :1- ££y 1, z = 0 . Figure the qualitative behaviour of the trajectories on the plane.

Exercise 7.8 Consider the system of ordinary differential equations ì \ ï y - y5 + y3 ()12+ z - y ()1 + z2 = 0 í (7.16) ï \ 4 2 2 4 2 î z + z ()14+ y - 2 y ()z + y - y + 32¤ = 0. Show that these equations generate a dissipative dynamical system in R2 . Verify that the set Myz= {(), : zy= 2 , y Î}R is invariant and exponentially attracting. Using Theorem 7.4, prove that the global attractor A of problem (7.16) has the form Ayz= {}(), : zy= 2, -1££y 1 . Hint: Consider the variable wzy= - 2 instead of the variable z . 52 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C h § 8 Finite Dimensionality a p of Invariant Sets t e r 1 Finite dimensionality is an important property of the global attractor which can be established in many situations interesting for applications. There are several approaches to the proof of this property. The simplest of them seems to be the one based on Ladyzhenskaya’s theorem on the finite dimensionality of the invariant set. However, it should be kept in mind that the estimates of dimension based on La- dyzhenskaya’s theorem usually turn out to be too overstated. Stronger estimates can be obtained on the basis of the approaches developed in the books by A. V. Babin and M. I. Vishik, and by R. Temam (see the references at the end of the chapter). Let MX be a compact set in a metric space . Then its fractal dimension is defined by (), e lnnM dimf M = lim , e ® 0 ln()1¤ e where nM(), e is the minimal number of closed balls of the radius e which cover the set M . Let us illustrate this definition with the following examples.

Example 8.1 Let Ml be a segment of the length . It is evident that

l - 1 ££nM(), e l + 1 . 2e 2 e Therefore, l - 2 e l + 2 e ln1 + ln ££lnnM(), e ln1 + ln . e 2 e 2

Hence, dimf M = 1 , i.e. the fractal dimension coincides with the value of the standard geometric dimension.

Example 8.2 Let M be the Cantor set obtained from the segment []01, by the sequentual removal of the centre thirds. First we remove all the points between 13¤ and 23¤ . Then the centre thirds ()19¤ , 29¤ and ()79¤ , 89¤ of the two remaining segments []013, ¤ and []23¤ , 1 are deleted. After that the centre parts ()127¤ , 227¤ , ()727¤ , 827¤ , ()19¤ 27, 20¤ 27 and ()25¤ 27, 26¤ 27 of the four remaining segments []019, ¤ , []29¤ , 13¤ , []23¤ , 79¤ and []89¤ , 1 , respectively, are deleted. If we continue this process to infinity, we obtain the Cantor set M . Let us calculate its fractal dimension. First of all it should be noted that Finite Dimensionality of Invariant Sets 53

¥ = MJÇ k , k = 0 = [], = [], ¤ È []¤ , J0 01, J1 013 23 1 , = [], ¤ È []¤ , ¤ È []¤ , ¤ È []¤ , J2 019 29 13 23 79 89 1 k and so on. Each set Jk can be considered as a union of 2 segments of the length 3-k . In particular, the cardinality of the covering of the set M with the segment of the length 3-k equals to 2k . Therefore, k ln2 ln2 dimf M ==lim . k ® ¥ ln()23× k ln3 Thus, the fractal dimension of the Cantor set is not an integer (if a set possesses this property, it is called fractal).

It should be noted that the fractal dimension is often referred to as the metric order of a compact. This notion was first introduced by L. S. Pontryagin and L. G. Shnirel- man in 1932. It can be shown that any compact set with the finite fractal dimension is homeomorphic to a subset of the space Rd when d > 0 is large enough. To obtain the estimates of the fractal dimension the following simple assertion is useful.

Lemma 8.1. The following equality holds: (), e lnNM dimf M = lim , e ® 0 ln()1¤ e where NM(), e is the cardinality of the minimal covering of the compact M with closed sets diameter of which does not exceed 2 e (the diameter of a set XdX is defined by the value ()= sup {}xy- : xy, ÎX ).

Proof. It is evident that NM(), e £ nM(), e . Since any set of the diameter d lies in a ball of the radius dnM , we have that (), 2 e £ NM(), e . These two inequalities provide us with the assertion of the lemma.

All the sets are expected to be compact in Exercises 8.1–8.4 given below.

Í £ Exercise 8.1 Prove that if M1 M2 , then dimf M1 dimf M2 . ()È £ {}; Exercise 8.2 Verify that dimf M1 M2 max dimf M1 dimf M2 . ´ Exercise 8.3 Assume that M1 M2 is a direct product of two sets. Then ()´ £ + dimf M1 M2 dimf M1 dimf M2 . 54 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C h Exercise 8.4 Let g be a Lipschitzian mapping of one metric space into a ()£ p another. Then dimf gM dimf M . t e r The notion of the dimension by Hausdorff is frequently used in the theory of dynamical systems along with the fractal dimension. This notion can be defined as follows. 1 Let MXd be a compact set in . For positive and e we introduce the value ()d m ()Md,,e = inf å rj , where the infimum is taken over all the coverings of the set M with the balls of the £me (),,e radius rj . It is evident that Md is a monotone function with respect toe . Therefore, there exists m ()Md, ==lim m()Md,,e sup m ()Md,,e . e ® 0 e > 0 The Hausdorff dimension of the set M is defined by the value

dimH M = inf {}d : m ()Md, = 0 .

Exercise 8.5 Show that the Hausdorff dimension does not exceed the fractal one.

Exercise 8.6 Show that the fractal dimension coincides with the Hausdorff one in Example 8.1, the same is true for Example 8.2. ¥ Exercise 8.7 Assume that Ma= {} Ì , where a monotonically n n = 1 R n tends to zero. Prove that dimH M =m0 (Hint: ()£Md,,e £ d + -dn ££e an +1 n 2 when an +1 an ). = {}¤ ¥ Ì = ¤ Exercise 8.8 Let M 1 n n = 1 R . Show that dimf M 12 . æ Hint:nnM<<(), e n ++1 1 when 1 £ e< è ()en +1 ()n +1 ()n +2 ö < 1 . nn()+1 ø

ìü¥ Exercise 8.9 Let M = íý1 Ì R . Prove that dim M = 1 . lnn f îþn = 2

Exercise 8.10 Find the fractal and Hausdorff dimensions of the global minimal attractor of the dynamical system in R generated by the differential equation \ y + y sin1 = 0 . y

The facts presented in Exercises 8.7–8.9 show that the notions of the fractal and Hausdorff dimensions do not coincide. The result of Exercise 8.5 enables us to restrict ourselves to the estimates of the fractal dimension when proving the finite dimensionality of a set. Finite Dimensionality of Invariant Sets 55

The main assertion of this section is the following variant of Ladyzhenskaya’s theorem. It will be used below in the proof of the finite dimensionality of global attractors of a number of infinite-dimensional systems generated by partial differential equations.

Theorem 8.1. Assume that MHVis a compact set in a Hilbert space . Let be a continuous mapping in HVM such that ()É M. Assume that there exists a finite- dimensional projector PH in the space such that ()- £ - , Î PVv1 Vv2 lv1 v2 ,,,v1 v2 M , (8.1) ()- ()d- £ - , Î 1 P Vv1 Vv2 v1 v2 ,,,v1 v2 M , (8.2) where d < 1 . We also assume that l ³ 1 - d . Then the compact M possesses a finite fractal dimension and 9 l 2 -1 dim M £ dim P ××ln ln . (8.3) f 1- d 1+ d

We remind that a projector in a space HP is defined as a bounded operator with the property P2 = P . A projector PPH is said to be finite-dimensional if the image is a finite-dimensional subspace. The dimension of a projector P is defined as a number dimP º dim PH . The following lemmata are used in the proof of Theorem 8.1.

Lemma 8.2. d Let BR be a ball of the radius R in R . Then d (), e ££(), e æö+ 2R NBR nBR èø1 e . (8.4) Proof. Estimate (8.4) is self-evident when e ³eR . Assume that < R . Let {}x ,,¼x x -ex > ¹ 1 l be a maximal set in BR with the property i j , ij . Îx By virtue of its maximality for every xBR there exists i such that -ex £ (), e £ x i . Hence, nBR l . It is clear that ()x Ì ()x ÇÆ()()x = ¹ Be¤ 2 i BR + e¤ 2 ,,Be¤ 2 i Be¤ 2 j ij. ()x x Here Br is a ball of the radius r centred at . Therefore, l ()= ()()x £ () lBVol e¤ 2 å Vol Be¤ 2 i Vol BR + e¤ 2 . i = 1 This implies the assertion of the lemma.

Exercise 8.11 Show that d (), e ³ æöR = nBR èøe , dimf BR d . 56 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C h Lemma 8.3. a p Let be a closed subset in H such that equations (8.1) and (8.2) hold t for all its elements. Then for any q >e0 and > 0 the following esti- e r mate holds: n 1 (), e ()+ d £ æö+ 4 l (), e NV q èø1 q N , (8.5) where nP= dim is the dimension of the projector P .

Proof. {} Let i be a minimal covering of the set with its closed subsets the diameter of which does not exceed 2 e . Equation (8.1) implies that in PH there e Ì exist balls Bi with radius 2 l such that PV i Bi . By virtue of Lemma 8.1 N {}i there exists a covering Bij j = 1 of the set PV i with the balls of the diameter e £ ()+ ()¤ n 2 q , where Ni 14lq . Therefore, the collection ìü íýG ==B + ()1- P V : i 12,,¼ ,N (), e , j =12,,¼ ,N îþij ij i i is a covering of the set V . Here the sum of two sets AB and is defined by the equality AB+ = {}ab+ : aAÎ , bBÎ . It is evident that £ + ()- diam Gij diamBij diam 1 P V i . ()- £ de £ Equation (8.2) implies that diam1 P V i 2 . Therefore, diamGij £ 2 ()eq + d . Hence, estimate (8.5) is valid. Lemma 8.3 is proved.

Let us return to the proof of Theorem 8.1. Since MVMÌ , Lemma 8.3 gives us that n (), e ()+ d £ (), e × æö+ 4 l NM q NM èø1 q . It follows that nm (), ()+ d m £ (), × æö+ 4 l , = ,,¼ NMq NM 1 èø1 q m 12 . We choose qmm and = ()e such that d + q

Obviously, the choice of m ()e can be made to fulfil the condition lne m ()e £ + 1 . ln()q + d Thus, æö4 l -1 dim M £ n ln 1 + ln1 . f èøq q + d

By taking q = 12¤ ()1- d we obtain estimate (8.3). Theorem 8.1 is proved.

Exercise 8.12 Assume that the hypotheses of Theorem 8.1 hold and l < 1- d . Prove that dimf M = 0 .

Of course, in the proof of Theorem 8.1 a principal role is played by equations (8.1) and (8.2). Roughly speaking, they mean that the mapping V squeezes sets along the space ()1- P HPH while it does not stretch them too much along . Negative invariance of MMV gives us that Ì kM for all k = 12,,¼ . Therefore, the set M should be initially squeezed. This property is expressed by the assertion of its finite dimensionality. As to positively invariant sets, their finite dimensionality is not guaranteed by conditions (8.1) and (8.2). However, as the next theorem states, they are attracted to finite-dimensional compacts at an exponential velocity.

Theorem 8.2. Let VM be a continuous mapping defined on a compact set in a Hil-Hil- bert space HVMM such that Ì . Assume that there exists a finite-dimensional projector P such that equations (8.1) and (8.2) hold with 0 <

Proof. The pair ()MV, k is a discrete dynamical system. Since M is compact, Theo- rem 5.1 gives us that there exists a global attractor M = VkM with the pro- 0 Çk ³ 0 = perties VM0 M0 and ()k , º {}()k , Î ® hVMM0 supdist V yM0 : yM 0 .(8.8) 58 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C j h We construct a set Aq as an extension of M0 . Let Ej be a maximal set in V M pos- a (), ³ q j , Î ¹ sessing the property dist ab for ab Ej , ab . The existence of such p j t a set follows from the compactness of V M . It is obvious that e r º = æö, 1 q j £ æöj , 1 q j Lj Card Ej NEèøj NVèøM . 1 3 3 - Lemma 8.3 with ==Mq,qd- , and e= ()q13¤ j 1 gives us that

æöæön æö- - NVjM, 1 q j £ 1 + 4 l NVj 1M, 1 q j 1 èø3 èøqd- èø3 with qd> . Hereinafter nP= dim . Therefore, æö4 l nj æö L ºqdCard E £ 1 + NM, 1 , > . (8.9) j j èøqd- èø3 Let us prove that the set = È {}È {}k ==,,¼ , ,,,¼ Aq M0 V Ej : j 12 k 012 (8.10) Ì k Ì possesses the properties required. It is evident that VAq Aq . Since V Ej Ì Vkj+ M , by virtue of (8.8) all the limit points of the set {}k ==,,¼ , ,,,¼ È V Ej : j 12 k 012

lie in M0 . Thus, Aq is a closed subset in M . The evident inequality ()k , ££()qk , k hVMAq hVMEk (8.11) implies (8.6). Here and below hXY(), = sup {}dist ()xY, : xXÎ . Let us prove (8.7). It is clear that = È {}È {}= ,,¼ Aq VAq Ej : j 12 .(8.12) {} Let Fi be a minimal covering of the set Aq with the closed sets the diameter e {} of which is not greater than 2 . By virtue of Lemma 8.3 there exists a covering Gi of the set VAq with closed subsets of the diameter 2 e ()q + d . The cardinality of this covering can be estimated as follows n ()e,,d º (), e ()+ d £ æö+ l (), e N q NVA q q èø14q NAq .(8.13) {} Using the covering Gi , we can construct a covering of the same cardinality of the (), e ()+ d e ()+ d set VAq with the balls Bxi 2 q of the radius 2 q centered at the , = ,,¼ , ()e,,d points xi i 12 N q . We increase the radius of every ball up to the value 2 e ()q ++dg . The parameter g> 0 will be chosen below. Thus, we consider the covering {}(), e ()++dg, = ,,¼ , ()e,,d Bxi 2 q i 12 N q

of the set VAq . It is evident that every point xVAÎ q belongs to this covering together with the ball Bx(), 2 ge . If j ³ 2 , the inequalities Finite Dimensionality of Invariant Sets 59

(), ££()j , ()j , hEj VAq hVMVAq hVMVEj -1 hold. By virtue of equation (8.11) with the help of (8.1) and (8.2) we have that ()j , ££()+ d ()j -1 , ()q+ d j -1 hVMVEj -1 l hV MEj -1 l . (), £ ge ge £ ()q+ d j -1 Therefore, hEj VAq 2 , provided 2 l , i.e. if l+ d ln 2 ge j ³ j º 2 + . 0 1 lnq Here []z is an integer part of the number z . Consequently, ìüN ()e,,q d È Ì (), e ()++dg VAq íýÈ Ej È Bxi 2 q . îþ³ i = 1 jj0 Therefore, equation (8.12) gives us that - j0 1 (), ex £ ()e,,d + NAq N q å Card Ej , j = 0 where x = 2 ()q ++dg . Using (8.9) and (8.13) we find that - j0 1 æö æönj NA()h, ex £ NA(), e + NM, 1 1 + 4 l q q èø3 å èøqd- j = 1 n qd>h= æö+ 4 l for . Here and further èø1 q . Since 1 lne j £ + Cl(),,,dgq, 0 1 lnq it is easy to find that - j0 1 æönj æöa n æö 1 + 4 l £ab × 1 for= × ln 1 + 4 l , å èøqd- èøe ln()1¤ q èøqd- j = 0 where the constant b >e0 does not depend on (its value is unessential further). Therefore, -a NA()hq, ex £ NA()beq, ex + . If we take ex= m - 1 , then after iterations we get

m m - 1 -()am - 1 NA()hq, x £ NA()bxq, x + £

m - 1 - £ hm ()bh, + m - 1 ()xa h i NAq 1 å . i = 0 60 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C d Îqd(), ¤ Î (), Îg(), ¤d- > h Let us fix 012 , 1 and q 012 and choose 0 such a a that x =x2 ()q ++dg< 1 and h ¹ 1 . Then summarizing the geometric progres- p t sion we obtain e x-a m - hm r ()h, xm ££m ()b, + NAq NAq 1 -a 1 x - h æöb b -a £ hm NA(), 1 + + x m . (8.14) èøq x-a - h x-a - h Let e > 0 be small enough and ln()1¤ e m ()e = 1 + , ln()1¤ x where, as mentioned above, []z is an integer part of the number z . Since ex£ m()e , equation (8.14) gives us that æönm()e æöm ()e NA(), e ££NA(), xm a 1 + 4 l + a 1 , q q 1 èøq 2 èøxa e where a1 and a2 are positive numbers which do not depend on . Therefore, lnN ()A , e q dimf Aq = lim £ e ® 0 1 lne ()e ìünm m £ m 1 ()+ 4 e + æö1 limlim íýa1 1 a2 èøa . e ® 0 1 m ® ¥ m q lne îþx Simple calculations give us that ìün £ 1 × æöæö+ 4 e , x-a dimf Aq ln íýmax èøèø1 . 1 îþq lnx This easily implies estimate (8.7). Thus, Theorem 8.2 is proved.

Exercise 8.13 Show that for dq<<12¤ formula (8.7) for the dimension of the set Aq can be rewritten in the form æö+ 4 l lnèø1 qd- dimf Aq = dim P × . (8.15) ln1 2 q

If the hypotheses of Theorem 8.2 hold, then the discrete dynamical system ()MV, k

possesses a finite-dimensional global attractor M0 . This attractor uniformly attracts all the trajectories of the system. Unfortunately, the speed of its convergence to the attractor cannot be estimated in general. This speed can appear to be small. However, Theorem 8.2 implies that the global attractor is contained in a finite-dimensional po- Existence and Properties of Attractors … 61 sitively invariant set possessing the property of uniform exponential attraction. From the applied point of view the most interesting corollary of this fact is that the dynamics of a system becomes finite-dimensional exponentially fast independent of the speed of convergence of the trajectories to the global attractor. Moreover, the reduction principle (see Theorem 7.4) is applicable in this case. Thus, finite-dimensional invariant exponentially attracting sets can be used to describe the qualitative behaviour of infinite-dimensional systems. These sets are frequently referred to as inertial sets, or fractal exponential attractors. In some cases they turn out to be surfaces in the phase space. In contrast with the global attractor, the inertial set of a dynamical system can not be uniquely determined. The construction in the proof of Theorem 8.2 shows it.

§ 9 Existence and Properties of Attractors of a Class of Infinite-Dimensional Dissipative Systems

The considerations given in the previous sections are mainly of general character. They are related to a dissipative dynamical system of the generic structure. There- with, we inevitably make additional assumptions on the behaviour of trajectories of these systems (e.g., the asymptotic compactness, the existence of a Lyapunov function, the squeezing property along a subspace, etc.). Thereby it is natural to ask what properties of the original objects of a particular dynamical system guarantee the fulfilment of the assumptions mentioned above. In this section we discuss this question in terms of the dynamical system generated by a differential equation of the form

d yAy+ ==By(), y y (9.1) dt t = 0 0 in a separable Hilbert space H , where A is a linear operator and B is a nonlinear mapping which is coordinated with A in some sense. Our main goal is to demonstrate the generic line of arguments as well as to describe those properties of the operators A and B which provide the applicability of general theorems proved in the previous sections. The main attention is paid to the questions of existence and finite dimensionality of a global attractor. Nowadays the presented line of arguments (or a modification of it) is one of the main components of a great number of works on global attractors. It is assumed below that the following conditions are fulfilled. 62 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C (A) h There exists a strongly continuous semigroup St of continuous map- a pings in H such that yt()= S y is a solution to problem (9.1) in the p t 0 t sense that the following identity holds: e r t S y = T y + Gt(), y º T y + T BS()ty ,(9.2) 1 t 0 t 0 0 t 0 ò t -t t 0 d 0 = ()- where Tt exp At (see condition (B) below). The semigroup St is dissipative, i.e. there exists R > 0 such that for any B from the collec- BH() H < tion of all bounded subsets of the space the estimate St y < Î ³ () R holds when yB and tt0 B . We also assume that the set g+()= ÎBH() B Èt ³ 0 St B is bounded for any B . = ()- (B) The linear closed operator AT generates a semigroup t exp At £ ()w w which admits the estimate Tt L1exp t (L1 and are some constants). There exists a sequence of finite-dimensional projectors {} Pn which strongly converges to the identity operator such that Ì 1)AP commutes with n , i.e. Pn AAPn for any n ; ()- £ ()-e > 2) there exists n0 such that Tt 1 Pn L2 exp t for nn0 , e > where , L 2 0 ; = -1()- ® ® ¥ 3)rn A 1 Pn 0 asn . (C) For any R¢ > 0 the nonlinear operator Bu() possesses the properties: ()- ()£ ()¢ - £ ¢ = , 1)Bu1 Bu2 C1 R u1 u2 if ui R , i 12 ; ³ £ ¢, = , s > 2) for nn0 , ui R i 12 , and for some 0 the following equations hold: s()- ()£ ()¢ A 1 Pn Bu1 C2 R , s()- ()()- ()£ ()¢ - A 1 Pn Bu1 Bu2 C3 R u1 u2 s ()- (the existence of the operator A 1 Pn follows from (B2)).

It should be noted that although conditions (A)–(C) seem a little too lengthy, they are valid for a class of problems of the theory of nonlinear oscillations as well as for a number of systems generated by parabolic partial differential equations. The following assertion should be mainly interpreted as a principal result which testifies to the fact that the asymptotic behaviour of the system is determined by a finite set of parameters.

Theorem 9.1.

If conditions (A)–(C) are fulfilled, then the semigroup St possesses a compact global attractor A . The attractor has a finite fractal dimension which can be estimated as follows: A = ()+ ()+ ()e-1 dimf a1 1 ln Pn L1 L2 1 DR dimPn , (9.3) Existence and Properties of Attractors … 63

()= w+ () ³ where DR L1C1 R and nn0 is determined from the condition - s £ ()[]() 1 × {}- ()()e+ -1 rn a2 DR L1 L2 C3 R exp a3 DR 1 lnL2 . (9.4) ,, Here a1 a2 and a3 are some absolute constantsconstants..

When proving the theorem, we mainly rely on decomposition (9.2) and the lemmata below.

Lemma 9.1. ³ = + Let Kn be a set of elements which for some t 0 have the form vu0 + ()- (), Î H £ 1 Pn Gt u, where u0 Pn , u0 c0 R with the constant c0 de- £ = ,,¼ termined by the condition Pn c0 for n 12 . Here the value Gt(), u is the same as in (9.2) with the element u Î H being such that £ ³ H St uR for all t 0 . Then the set Kn is precompact in for ³ nn0 .

Proof. Properties (B2) and (C2) imply that t C ()R L s()- ()£ -e ()t - t s()- ()t£ 2 2 A 1 Pn Gt, u L2 òe A 1 Pn BSt u d e 0

when St uR£ for t ³ 0 . Therefore, the set {}= ()- ()> v : v 1 Pn Gt, u , t 0 ,(9.5) £ ³ ()s ()- H where St uR for all t 0 , is bounded in the space DA 1 Pn with the norm As . . The symbol denotes the restriction of an operator on a subspace. However, property (B3) implies that -1()- - -1()- = lim Pm A 1 Pn A 1 Pn 0 . m ® ¥ -1 ()- H ()s ()- H Therefore, the operator A 1 Pn is compact. Hence, DA 1 Pn ()- H is compactly embedded into 1 Pn . It means that the set (9.5) is precom- ()- H pact in 1 Pn . This implies the precompactness of Kn . Lemma 9.2. There exists a compact set K in the space H such that -e ()tt- ()º {}()Î £ 0 hSt B , K sup dist St y , K : yB L2 Re (9.6) Ì H ³ º () for any bounded set B and tt0 t0 B .

Proof. Î H £ ³ Let uB , where B is a bounded set in . Then St uR for tt0 = () = t0 B . By virtue of (9.2) we have that 64 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C = ()- + () h St u 1 Pn Ttt- St uWn t , t0 , u , a 0 0 p where t e W ()t , t , u = P S u + ()1- P Gt()- t, S u . r n 0 n t n 0 t0 ()Î ³ 1 It is evident that Wn t , t0 , u Kn for tt0 . Therefore, -e ()- tt0 dist()S u , K ££() 1- P T - S uLRe . t n n tt0 t0 2 = [] [] H This implies (9.6) with KKn , where Kn is the closure in of the set Kn described in Lemma 9.1. = [] Exercise 9.1 Show that KKn lies in the set = { + Î H Î ()- H, Kn, s v1 v2 : v1 Pn , v2 1 Pn £ s £ } v1 C1 , A v2 C2 , (9.7)

where C1 and C2 are some constants. ()H In particular, Lemma 9.2 means that the system , St is asymptotically compact. Therefore, we can use Theorem 5.1 (see also Exercise 5.3) to guarantee the exis- A = [] tence of the global attractor lying in KKn . Let us use Theorem 8.1 to prove the finite dimensionality of the attractor. Veri- fication of the hypotheses of the theorem is based on the following assertion.

Lemma 9.3. £ ³ = , Let St ui R , t 0, i 12 . Then - £ ()() - St u1 St u2 L1 exp DRt u1 u2 (9.8) and () æös L C R ()1- P ()S u - S u £ L e-e t × 1 + C r 1 3 ea t u - u (9.9) n t 1 t 2 2 èø0 n DR() 1 2 ³ae= + () for nn0 and DR .

Proof. Decomposition (9.2) and condition (C1) imply that æöt - £ ç÷- + () -w t -twt St u1 St u2 L1 ç÷u1 u2 C1 R ò e St u1 St u2 d e . èø0 With the help of Gronwall’s lemma we obtain (9.8). To prove (9.9) it should be kept in mind that decomposition (9.2) and equa- ³ tions (B2) and (C2) imply that for nn0 Existence and Properties of Attractors … 65

æ ()- ()- £ -e t ç - + 1 Pn St u1 St u2 L2 e ç u1 u2 è t ö + s () et -t÷ (9.10) C0 rn C3 R òe St u1 St u2 d ÷ . 0 ø -s ()- £ s Here the inequality A 1 Pn C0 rn is used . If we put (9.8) in the right- hand side of formula (9.10), we obtain estimate (9.9).

The following simple argument completes the proof of Theorem 9.1. Let us fix <

Exercise 9.2 Prove that the global attractor A of problem (9.1) is stable (Hint: verify that the hypotheses of Theorem 7.1 hold).

Properties (A)–(C) also enable us to prove that the system generated by equation H (9.1) possesses an inertial set. A compact set Aexp in the phase space is said to be an inertial set (or a fractal exponential attractor) if it is positively invariant ()Ì ()< ¥ St Aexp Aexp , its fractal dimension is finite dimf Aexp and it possesses the property -g ()tt- (), º {}()Î £ 0 hSt BAexp sup dist St y , Aexp : yB CB e (9.11) Ì H ³³() g for any bounded set B and for tt0 t0 B , where CB and are positive numbers. (The importance of this notion for the theory of infinite-dimensional dynamical systems has been discussed at the end of Section 8).

Lemma 9.4. Assume that properties (A)–(C) hold. Then the dynamical system ()H , St generated by equation (9.1) possesses the following properties: 1) there exist a compact positively invariant set K and constants C , g > 0 such that -g ()tt- {}(), Î £ B sup dist St yK: yB Ce (9.12) H ³ > for any bounded set B in and for ttB 0 ; 66 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C O D h 2) there exist a vicinity of the compact K and numbers 1 and a a > 0 such that p 1 t a t - £ D 1 - e St y1 St y2 1 e y1 y2 , (9.13) r ³ provided that for all t 0 the semitrajectories St yi lie in the clo- 1 sure []O of the set O ; {} 3) there exist a sequence of finite-dimensional projectors Pn in the H Dab > space , constants 2 , 2 , 0 , and a sequence of positive {}r ® ¥ numbers n tending to zero as n such that a t ()- ()- £ D -b t ()+ r 2 - 1 Pn St y1 St y2 2 e 1 n e y1 y2 (9.14) Î for any y1, y2 K .

Proof. Let K be a compact set from Lemma 9.2. Let * = g+ ()º K K È St K . t ³ 0 * Ì * = * = It is clear that St K K and equation (9.12) holds for KK with CL2 R ge= * {} and . Let us prove that K is a compact set. Let zn be a sequence of elements of K* = g+()K . Then z = S y for some t > 0 and y Î K . n tn n n n If there exists an infinitely increasing subsequence {}t , then equation (9.6) nk gives us that æö, = limdist èøSt yn K 0 . k ® ¥ nk k {} Ì * {} Therefore, the sequence zn possesses a limit point in KK . If tn is a bounded sequence, then by virtue of the compactness of K there exist a number t > 0 , an element yKÎ and a sequence {}n such that y ® y and 0 k nk t ® t . Therewith nk - £ - + - St yn St y St ySt y St yn St y . nk k 0 n 0 nk k nk The first term in the right-hand side of this inequality evidently tends to zero. As for the second term, our argument is the same as in the proof of formula (9.8). We use the boundedness of the set g+()K (see property (A)) and properties (B) and (C2) to obtain the estimate C t - £ K - Î St y1 St y2 Ce y1 y2 , y1, y2 K . (9.15) It follows that - = limSt yn St y 0 . k ® ¥ nk k nk Existence and Properties of Attractors … 67

Therefore, ®gÎ * = +() St yn St yK K . nk k 0 The closedness of the set K* can be established with the help of similar arguments. Thus, property (9.12) is proved for KK= * . Now we suppose that KS= K* , where t is chosen such that yR< for all yKÎ . It is obvious t0 0 that K is a compact positively invariant set. As it is proved above, it is easy to

find the estimate of form (9.15) for all y1 and y2 from an arbitrary bounded set B . Here an important role is played by the boundedness of the set g+()B (see ÎBH() = property (A)). Therefore, for any B there exists a constant C0 = ()* > CB, K , t0 0 such that S y - S y £ C y - y , y , y Î BKÈ * . t0 1 t0 2 0 1 2 1 2 Hence, for yBÎ we have that * * dist ()S y , K = dist ()S × S - y , S K £ C dist ()S - y, K t t0 tt0 t0 0 tt0 > for tt0 . This implies estimate (9.12) with the constant CK depending on and Bt . However, if we change the moment B in equation (9.12), we can pre- sume that, for example, C =ge1 . Therewith = . Thus, the first assertion of the lemma is proved. Since the set Kz lies in the ball of dissipativity {}ÎH : zR£ , estimates (9.13) and (9.14) follow from Lemma 9.3. Moreover, D ==a () D =a =e + () 1 L1, 1 DR, 2 L2 , 2 DR , bge==r = s ()× ()-1 , n C0 rn C3 R DR .(9.16) Thus, Lemma 9.4 is proved.

Lemma 9.4 along with the theorem given below enables us to verify the existence of an inertial set for the dynamical system generated by equation (9.1).

Theorem 9.2. HH() Let the phase space of a dynamical system , St be a Hilbert space. Assume that in H there exists a compact positively invariant set K possessing properties (9.12)–(9.14). Then for any n > ln2 there exists an n ()H inertial set Aexp of the dynamical system , St such that ìüga+ ()n £ ()n × - gæö- 1 ()- hSt B , Aexp CB, exp íýèø1 nga++ ttB (9.17) îþ1 ³ (), = { (), for any bounded set Btt and B . Here, as above, hXY sup dist xY: xXÎ}. Moreover, n £ × ()+ dimf Aexp C0 1 ln Pn dim Pn , (9.18) 68 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C h where the number n is determined from the condition a a p 2 ìüa t r £ ()D b × -n 2 n 4 2 exp íý (9.19) e îþb r and constant C does not depend on n and n . 1 and constant 0 does not depend on and . The proof of the theorem is based on the following preliminary assertions.

Lemma 9.5. () Let K , St be a dynamical system, its phase space being a compact in H (), Î a Hilbert space . Assume that for all y1 y2 K equations (9.13) and (9.14) are valid. Then for any n > ln2 there exists an inertial set n () Aexp of the system K , St such that ()n = {}(), n Î £ -n t hSt K , Aexp sup dist St yAexp : yK Cn e . (9.20) n Moreover, estimate (9.18) holds for the value dimf Aexp .

Proof. We use Theorem 8.2 with MK= , VS=d , and = 1 e-n , where t and t0 2 0 n are chosen to fulfil -b t d a t D e 0 == 1 e-n and re 2 0 £ 1 . 2 2 4 n =d= 1 -n In this case conditions (8.1) and (8.2) are valid for VSt with e and a t 0 2 = D 1 0 lPn 1 e . Therefore, there exists a bounded closed positively invariant set Aq with dq<<12¤ such that (see (8.6) and (8.15)) m m sup{}qdist ()V y , Aq : yKÎ £ , m = 1,, 2 ¼ (9.21) and -1 æö4 l dim A £ ln 1 + ln1 × dim P (9.22) f q èøqd- 2q n Assume that q ==2 d e-n and consider the set n = {}££ Aexp È St Aq :0 0 t0 . Here n = ln1 > ln2 . It is easy to see that q n £ + n dimf Aexp 1 dimf Aexp . Therefore, equations (9.20) and (9.18) follow from (9.21) and (9.20) after some simple calculations.

Lemma 9.6. HH() Assume that in the phase space of a dynamical system , St Ì there exist compact sets KK and 0 such that (a) K0 K ; (b) properties Existence and Properties of Attractors … 69

(9.12) and (9.13) are valid for KK ; and (c) the set 0 possesses the property -g t ()£ 0 hSt K , K0 Ce , (9.23) where hXY(), = sup {}dist ()xY, : xXÎ . Then for any bounded set B Ì H ³ and ttB the following inequality holds ìügg ()£ - 0 hSt B , K0 CB exp íýgg++a t . (9.24) îþ0 1

Proof. By virtue of (9.12) every bounded set B reaches the vicinity O in finite time and stays in it. Therefore, it is sufficient to prove the lemma for a set ÎBH() Ì []O ³ []O O B such that St B for t 0 , where denotes the closure of . Î Î Let k0 K0 and yB . Evidently, - £ - + - St yk0 S t S()1- t yS t k S t kk0 for any 0 ££ 1 and kKÎ . With the help of (9.13) we have that a t - £ D 1 - + - St yk0 1 e S()1- t yk S t kk0 . Therefore, for any 0 << 1 and kKÎ we have that a t ()££D 1 - + () distSt y , K0 1 e S()1- t yk dist S t k , K0 a t £ D 1 - + () 1 e S()1- t yk hS t K , K0 . If we take an infimum over kKÎ and a supremum over yBÎ , we find that a t ()£ D 1 ()+ () hSt B , K0 1 e hS()1- t B , K hS t K , K0 for all 0 << 1 . Hence, equations (9.12) and (9.23) give us that ()a g- ()1- t -g t ()£ 1 + 0 hSt B , K0 CB e CK e ³ggg= ()++a -1 for ttB . If we choose 0 1 , we obtain (9.24). Lemma 9.6 is proved.

= n If we now use Lemma 9.6 with K0 Aexp and estimate (9.20), we get equation (9.17). This completes the proof of Theorem 9.2.

Thus, by virtue of Lemma 9.4 and Theorem 9.2 the dynamical system ()H , S gene- n t rated by equation (9.1) possesses an inertial set Aexp for which equations (9.17)– (9.19) hold with relations (9.16). It should be noted that a slightly different approach to the construction of inertial sets is developed in the book by A. Eden, C. Foias, B. Nicolaenko, and R. Temam (see the list of references). This book contains further developments and applications of the theory of inertial sets. 70 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C h To conclude this section, we outline the results on the behaviour of the projec- a H tion onto the finite-dimensional subspace Pn of the trajectories of the system p ()H t , St generated by equation (9.1). e Assume that an element y belongs to the global attractor A of a dynamical r 0 ()H, g= {}() Î system St . Lemma 6.1 implies that there exists a trajectory yt, t R 1 A ()= lying in wholly such that y 0 y0 . Therewith the following assertion is valid. Lemma 9.7. ÎA Assume that properties (A)–(C) are fulfilled and let y0 . Then the following equation holds: 0 ()- = ()- ()t()t ³ 1 Pn y0 ò 1 Pn T-t By d , nn0 , (9.25) -¥ {}()t where y is a trajectory passing through y0 , the number n0 can be found from (B2) and the integral in (9.25) converges in the norm of the space H .

Proof. = ()- Since y0 St yt , equation (9.2) gives us that æö0 ()- = ()- ç÷()- + ()- ()t()t 1 Pn y0 1 Pn ç÷Tt yt ò 1 Pn T-t By d .(9.26) èø-t A trajectory in the attractor possesses the property yt() £ R , t Î ()-¥ , ¥ . Therefore, property (B2) implies that ()- ()- £ × -e t ()- ()()t £ -et 1 Pn Tt yt L2 Re and1 Pn T-t By L2 CR e . These estimates enable us to pass to the limit in (9.26) as t ® -¥ . Thereupon we obtain (9.25).

The following assertion is valid under the hypotheses of Theorem 9.1.

Theorem 9.3. ³ ³ There exists N0 n0 such that for all NN0 the following assertions are valid: () () 1) for any two trajectories y1 t and y2 t lying in the attractor of the ()= () system generated by equation (9.1) the equality PN y1 t PN y2 t Î ()º () for all t R implies that y1 t y2 t ; () () 2) for any two solutions u1 t and u2 t of the system (9.1) the equation ()()- () = limPN u1 t u2 t 0 (9.27) t ® -¥ ()- () ® ® ¥ implies that u1 t u2 t 0 as t . Existence and Properties of Attractors … 71

We can also obtain an upper estimate of the number N0 from the inequali- s -1 ty r £ a e ()L C ()R . N0 0 2 3 Proof. () Equation (9.25) implies that for any trajectory yi t lying in the attractor of system (9.1) the equation t ()- ()==()- ()t()t , 1 PN yi t ò 1 PN Tt - t Byi d , i 12 , -¥ ()= () holds. Therefore, if PN y1 t PN y2 t , then properties (B2), (B3), and (C2) give us that t ()- () £ s () -e ()t - t ()t -t()t y1 t y2 t C0 rN L2 C3 R ò e y1 y2 d . -¥ It follows that the estimate

t0 ()- () £ {}- e + ()- et ()t -t()t y1 t y2 t AN exp t AN tt0 ò e y1 y2 d -¥ ³ = s () ® -¥ holds for tt0 , where AN C0 rN L2 C3 R . If we tend t0 , we obtain the < e first assertion, provided AN . Now let us prove the second assertion of the theorem. Let a ()= ()()- () N t PN u1 t u2 t . Then ()-a() £ ()+ ()- ()()- () u1 t u2 t N t 1 PN u1 t u2 t . y ()=e- () Therefore, equation (9.10) for the function t u1 u2 exp t gives us that t y()£ a () e t + ()- ()+ yt()t t N t e L2 u1 0 u2 0 AN ò d . 0 This and Gronwall’s lemma imply that ()-a() £ ()+ ()-()e - ()- ()+ u1 t u2 t N t L2 exp AN t u1 0 u2 0

t + a ()t ()t-() e- ()- t AN ò N exp AN t d . 0 < e ()- () ® Therefore, if AN , then equation (9.27) gives us that u1 t u2 t 0 . Thus, the second assertion of Theorem 9.3 is proved. 72 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems

C {}= ,,¼ h Theorem 9.3 can be presented in another form. Let ek : k 1 dN be a basis a H ()= ()H = in the space PN . Let us define linear functionals lj u u , ej on , j p = , ¼, t 1 dN . Theorem 9.3 implies that the asymptotic behaviour of trajectories of e the system ()H , S is uniquely determined by its values on the functionals l . r t j {} Therefore, it is natural that the family of functionals lj is said to be the determin- 1 ing collection. At present some general approaches have been worked out which enable us to define whether a particular set of functionals is determining. Chapter 5 is devoted to the exposition of these approaches. It should be noted that for the first time Theorem 9.3 was proved for the two-dimensional Navier-Stokes system by C. Foias and D. Prodi (the second assertion) and by O. A. Ladyzhenskaya (the first assertion). Concluding the chapter, we would like to note that the list of references given below does not claim to be full. It contains only references to some monographs and reviews devoted to the developments of the questions touched on here and compris- ing intensive bibliography. References

1. BABIN A. V., VISHIK M. I. Attractors of evolution equations.–Amsterdam: North-Holland, 1992. 2. TEMAM R. Infinite-dimensional dynamical systems in Mechanics and Physics. –New York: Springer, 1988. 3. HALE J. K. Asymptotic behavior of dissipative systems. –Providence: Amer. Math. Soc., 1988. 4. EDEN A., FOIAS C., NICOLAENKO B., TEMAM R. Exponential attractors for dissipative evolution equations. –New York: Willey, 1994. 5. LADYZHENSKAYA O. A. On the determination of minimal global attractors for the Navier-Stokes equations and other partial differential equations // Russian Math. Surveys. –1987. –Vol. 42. –# 6. –P. 27–73. 6. KAPITANSKY L. V., KOSTIN I. N. Attractors of nonlinear evolution equations and their approximations // Leningrad Math. J. –1991. –Vol. 2. –P. 97–117. 7. CHUESHOV I. D. Global attractors for nonlinear problems of mathematical physics // Russian Math. Surveys. –1993. –Vol. 48. –# 3. –P. 133–161. 8. PLISS V. A. Integral sets of periodic systems of differential equations. –Moskow: Nauka, 1977 (in Russian). 9. HALE J. K. Theory of functional differential equations. –Berlin; Heidelberg; New York: Springer, 1977.

Chapter 2

Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

Contents

. . . . References ...... 145

In this chapter we study well-posedness and the asymptotic behaviour of solutions to a class of abstract nonlinear parabolic equations. A typical representative of this class is the nonlinear heat equation d ¶ u ¶2 u = n å + fx(), u ¶ t ¶ x2 i = 1 i considered in a bounded domain W of Rd with appropriate boundary conditions on the border ¶W . However, the class also contains a number of nonlinear partial differential equations arising in Mechanics and Physics that are interesting from the applied point of view. The main feature of this class of equations lies in the fact that the corresponding dynamical systems possess a compact absorbing set. The first three sections of this chapter are devoted to the questions of existence and uniqueness of solutions and a brief description of examples. They are independent of the results of Chapter 1. In the other sections containing the discussion of asymptotic properties of solutions we use general results on the existence and properties of global attractors proved in Chapter 1. In Sections 6 and 7 we present two quite simple infinite-dimensional systems for which the asymptotic behaviour of the trajectories can be explicitly described. In Section 8 we consider a class of systems generated by infinite-dimensional retarded equations. The list of references at the end of the chapter consists only of the books re- commended for further reading.

§ 1 Positive Operators with Discrete Spectrum

This section contains some auxiliary facts that play an important role in the subsequent considerations related to the study of the asymptotic properties of solutions to abstract semilinear parabolic equations. Assume that H is a separable Hilbert space with the inner product ().., and the norm . . Let ADA be a selfadjoint positive linear operator with the domain () . An operator A is said to have a discrete spectrum if in the space H there exists {} an orthonormal basis ek of the eigenvectors: ()d, = = l , = ,,¼ ek ej kj,Aek k ek , kj 12 , (1.1) such that

C 2 h Exercise 1.1 Let HL= ()01, and let A be an operator defined by the a = - ² () p equation Au u with the domain DA which consists of conti- t nuously differentiable functions ux() such that (a) u()0 ==u()1 0 , e 2 r (b) u¢()x is absolutely continuous and (c) u ² Î L ()01, . Show 2 that A is a positive operator with discrete spectrum. Find its eigenvectors and eigenvalues.

The above-mentioned structure of the operator A enables us to define an operator fA() for a wide class of functions f ()l defined on the positive semiaxis. It can be done by supposing that ¥ ¥ ìü ()() = = Î 2 []()l 2 < ¥ DfA íýhcå k ek H : å ck f k , îþ k = 1 k = 1 ¥ () = ()l Î ()() fAhcå k f k ek ,.(1.3)hDfA k = 1 In particular, one can define operators Aa with a ÎabR . For = - < 0 these operators are bounded. However, in this case it is also convenient to introduce the line- als DA()a if we regard DA()-b as a completion of the space H with respect to the norm A-b . .

-b Exercise 1.2 Show that the space -b =bDA() with > 0 can be identi- fied with the space of formal series åck ek such that ¥ 2 l-2 b < ¥ å ck k . k = 1

Exercise 1.3 Show that for any b Î R the operator Ab can be defined on every space DA()a as a bounded mapping from DA()a into DA()ab- such that b + b b b Ab DA()a = DA()ab- , A 1 2 = A 1 × A 2 . (1.4)

a Exercise 1.4 Show that for all a Î R the space a º DA() is a sepa- a a rable Hilbert space with the inner product ()uv, a = ()A uA, v a and the norm u a = A u . Exercise 1.5 The operator A with the domain 1 + s is a positive operator with discrete spectrum in each space s .

Exercise 1.6 Prove the continuity of the embedding of the space a into b for ab> , i.e. verify that a Ì b and u b £ Cua .

Exercise 1.7 Prove that a is dense in b for any ab> . Positive Operators with Discrete Spectrum 79

Exercise 1.8 Let f Îss for > 0 . Show that the linear functional Fg()º º ()fg, can be continuously extended from the space H to -s and ()fg, £ f s × g -s for any f Î s and g Î -s .

Exercise 1.9 Show that any continuous linear functional F on s has the form: Ff()= ()fg, , where g Î -s . Thus, -s is the space of continuous linear functionals on s .

The collection of Hilbert spaces with the properties mentioned in Exercises 1.7–1.9 is frequently called a scale of Hilbert spaces. The following assertion on the compactness of embedding is valid for the scale of spaces {}s .

Theorem 1.1.

Let s > s . TheThenn the space s is compactly embedded into s , i.e. 1 2 1 2 every sequence bounded in s is compact in s . 1 2 Proof. It is well known that every bounded set in a separable Hilbert space is weakly compact, i.e. it contains a weakly convergent sequence. Therefore, it is sufficient to prove that any sequence weakly tending to zero in s converges to zero with re- 1 spect to the norm of the space s . We remind that a sequence {}f in weakly 2 n s converges to an element f Î s if for all g Î s (), = (), lim fn g s fgs . n ® ¥ Let the sequence {}f be weakly convergent to zero in s and let n 1 £ = ,,¼ fn s C ,n 12 . (1.5) 1 Then for any N we have N - 1 ¥ 2 s 2 s 2 £ l 2 (), 2 + 1 l 1 (), 2 fn s å k fn ek ()s - s å k fn ek .(1.6) 2 2 1 2 k = 1 N kN= Here we applied the fact that for kN³ 2 s 1 2 s l 2 £ l 1 . k 2 ()s - s k l 1 2 N Equations (1.5) and (1.6) imply that N - 1 2 s -2 ()s - s 2 £ l 2 (), 2 + l 1 2 fn s å k fn ek C N . 2 k = 1 We fix e > 0 and choose N such that N - 1 2 s 2 £ l 2 (), 2 + e fn s å k fn ek .(1.7) 2 k = 1 80 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h Let us fix the number Nf . The weak convergence of n to zero gives us a lim()f , e = 0 ,.k = 12,,¼ ,N - 1 p ® ¥ n k t n e r Therefore, it follows from (1.7) that £ e 2 lim fn s . n ® ¥ 2 By virtue of the arbitrariness of e we have = limfn s 0 . n ® ¥ 2 Thus, Theorem 1.1 is proved.

()=ll()- l -1 ¹ Exercise 1.10 Show that the resolvent Rl A A , k , is a compact operator in each space s .

We point out several properties of the scale of spaces {}s that are important for further considerations.

Exercise 1.11 Show that in each space s the equation l = (), Î¥ -s¥<< Pl uueå k ek ,,u s k = 1 defines an orthoprojector onto the finite-dimensional subspace ge- {}, = ,,¼ , nerated by the set of elements ek k 12 l . Moreover, for each s we have - = limPl uus 0 . l ® ¥

Exercise 1.12 Using the Hölder inequality 1p 1q æöp æöq a b £ ç÷a ç÷b ,,,1 + 1 = 1 a , b > 0 å k k èøå k èøå k p q k k k k k prove the interpolation inequality q - q AquAu£ × u 1 ,,.0 ££q 1 uDAÎ ()

Exercise 1.13 Relying on the result of the previous exercise verify that s , s Î for any 1 2 R the following interpolation estimate holds: q 1- q u sq() £ u s u s , 1 2

where sq()= qs+ ()s1-q , 0 ££q 1 , and u Î ()s , s . 1 2 max 1 2 Prove the inequality q - 2 2 1- q 2 u sq() £ e u s + Cq e u s , 1 2 where 0 <

Equations (1.3) enable us to define an exponential operator exp()-tA , t ³ 0 , in the scale {}s . Some of its properties are given in exercises 1.14–1.17.

Exercise 1.14 For any a Î R and t > 0 the linear operator exp()-tA -tA maps a into Ç s and possesses the property e u a £ -t l s ³ 0 £ e 1 u a . Exercise 1.15 The following semigroup property holds: ()- × ()- = ()-()+ , ³ exp t1 A exp t2 A exp t1 t2 A ,.t1 t2 0

Exercise 1.16 For any u Îss and Î R the following equation is valid: -tA -t A lime ue- u s = 0 .(1.8) t ® t

Exercise 1.17 For any s ÎR the exponential operator e-tA defines a dissi- -tA pative compact dynamical system ()s , e . What can you say about its global attractor?

Let us introduce the following notations. Let Cab(), ; a be the space of strongly continuous functions on the segment []ab, with the values in a , i.e. they are con- a tinuous with respect to the norm . a = A . . In particular, Exercise 1.16 means -tA 1 that e uCÎ ()R+, a if u Î a . By C ()ab, ; a we denote the subspace of Cab(), ; a that consists of the functions ft() which possess strong (in a ) deri- k vatives f ¢()t lying in Ca(), b; a . The space C ()ab, ; a is defined similarly for any natural k . We remind that the strong derivative (in a ) of a function ft() at = Î a point tt0 is defined as an element v a such that 1 ()()+ - ()- = limft0 h ft0 v 0 . h ® 0 h a

Îs Exercise 1.18 Let u0 s for some . Show that -tA Î k ()d+¥, ; e u0 C a for all d >a0 , Î R , and k = 12,,¼ . Moreover, k d e-tAu = ()-A k e-tAu ,.k = 12,,¼ dtk 0 0

2 Let L ()ab, ; a be the space of functions on the segment []ab, with the values in a for which the integral b u 2 = ut()2 t 2 (), ; ò a d L ab a a ¥ exists. Let L ()ab, ; a be the space of essentially bounded functions on []ab, with the values in a and the norm 82 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C = a () h u ¥ (), ; ()a ess sup A ut . a L abDA tabÎ [], p t We consider the Cauchy problem e dy r + Ay = ft(),;tabÎ (), ya()= y , (1.9) dt 0 2 Î ()Î 2 (), ; where y a and ft L abFa - 12¤ . The weak solution (in a ) to this problem on the segment []ab, is defined as a function ()Î (), ; Ç 2 (), ; yt Ca b a L ab a + 12¤ (1.10) ¤ Î 2 (), ; such that dy dt L abFa- 12¤ and equalities (1.9) hold. Here the derivative y¢()t º dy¤ dt is considered in the generalized sense, i.e. it is defined by the equality b b j () ¢() =j- j¢() () Î ¥ (), ò t y t dt ò t ytdt ,,C0 ab a a ¥(), (), where C0 ab is the space of infinitely differentiable scalar functions on ab vanishing near the points ab and .

Exercise 1.19 Show that every weak solution to problem (1.9) possesses the property t t yt()2 + y ()t 2 dt £ y + f ()t 2 dt .(1.11) a ò a + 1 0 a ò a - 1 2 2 a a ()= () (Hint: first prove the analogue of formula (1.11) for yn t Pn yt , then use Exercise 1.11).

Exercise 1.20 Prove the theorem on the existence and uniqueness of weak solutions to problem (1.9). Show that a weak solution yt() to this problem can be represented in the form t ()= -()ta- A + -()t - t A ()tt yt e y0 òe f d .(1.12) a Î () Exercise 1.21 Let y0 a and let a function ft possess the property ()- () £ - q ft1 ft2 a Ct1 t2 for some 0 < q £ 1 . Then formula (1.12) gives us a solution to problem (1.9) belonging to the class ()[], ; Ç 1 ()] , ] ; Ç ()] , ] ; Cab a C ab a C ab a +1 .

Such a solution is said to be strong in a . Positive Operators with Discrete Spectrum 83

The following properties of the exponential operator e-tA play an important part in the further considerations.

Lemma 1.1.

Let QN be the orthoprojector onto the closure of the span of elements {}, ³ + = - = ,,,¼ ek kN1 in HP and let N IQN , N 012 . Then 1) for all hHÎb, ³ 0 and t Î R the following inequality holds: b l t b -tA £ l N A PN e h N e h ; (1.13) 2) for all hDAÎ ()b , t >ab0 and ³ the following estimate is valid: ab- æöab- ab- -t l Aa Q e-tAh £ + l e N +1 Ab h , (1.14) N èøt N +1 in the case ab- = 0 we supose that 00 = 0 in (1.14).

Proof. Estimate (1.13) follows from the equation N -2 t l b -tA 2 = l2 b k (), 2 A PN e h å k e hek . k = 1 In the proof of (1.14) we similarly have that ¥ a -tA 2 £ ()lab- -t l 2 l2 b (), 2 A QN e h max e å k hek . ll³ + N 1 kN= + 1 This gives us the inequality a -tA £ 1 ()mab- -m b A QN e h ab- max e A h . t m ³ l t N +1 Since max{}mg e-m ; m ³ 0 is attained when mg= , we have that

ì -l t ()l g e N + 1 , l ³ g ï N + 1 t if N + 1 t ; max ()mg e-m = í ml³ t N +1 ï gg -g, l < g î e if N + 1 t . Therefore, -l g -m g g N + 1 t max ()gm e £ ()+ ()l + t e . ml³ N 1 N +1 t This implies estimate (1.14). Lemma 1.1 is proved.

In particular, we note that it follows from (1.14) that ab- æöab- ab- -t l Aa e-tAh £ + l × e 1 Ab h ,.(1.15)ab³ èøt 1 84 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h Exercise 1.22 Using estimate (1.15) and the equation a p t t e-tAue- -sAu = - Ae-t A u t ,,,ts³ u Î e ò d b r s 2 prove that -tA -sA sq- e ue- u q £ Cqs, ts- u s ;,(1.16)ts, > 0

provided qs< £ 1 + q , where a constant Cqs, does not depend on ts and (cf. Exercise 1.16).

Exercise 1.23 Show that æöa a Aa e-tA £ e-a ,,.(1.17)t >a0 > 0 èøt

Lemma 1.2. ¥ Let ft()Î L ()R, ag- for 0 £ g < 1 . Then there exists a unique solution vt()Î C ()R, a to the nonhomogeneous equation v d + Av = ft(),,t Î R (1.18) dt

that is bounded in a on the whole axis. This solution can be represented in the form t -()- t vt()= ò e t A f ()tt d . (1.19) -¥

We understand the solution to equation (1.18) on the whole axis as a function vt()Î Î C ()R, a such that for any ab< the function vt() is a weak solution [], = () (ina-12¤ ) to problem (1.9) on the segment ab with y0 va .

Proof. If there exist two bounded solutions to problem (1.18), then their difference wt() is a solution to the homogeneous equation. Therefore, wt()= = {}-()- () ³ exp tt0 A wt0 for tt0 and for any t0 . Hence, -()ltt- -()ltt- a () £ 0 1 a () £ 0 1 A wt e A wt0 Ce . ® -¥ ()= If we tend t0 here, then we obtain that wt 0 . Thus, the bounded solution to problem (1.18) is unique. Let us prove that the function vt() defined by formula (1.19) is the required solution. Equation (1.15) implies that g g g -()lt - t a -()t - t A £ æö + l 1 ag- ()t A e èø- t 1 e ess sup A f t t Î R for t > t and 0 <

1+ k Aavt() £ ess sup Aag- f ()t , l1- g t ÎR 1 where k =g0 for = 0 and ¥ k = gg òs-g e-sds for0 <

t0 This also implies (see Exercise 1.18) that vt() is a solution to equation (1.18). Lemma 1.2 is proved.

§ 2 Semilinear Parabolic Equations in Hilbert Space

In this section we prove theorems on the existence and uniqueness of solutions to an evolutionary differential equation in a separable Hilbert space H of the form u d + Au = Bu(), t,u = u , (2.1) dt ts= 0 where AB is a positive operator with discrete spectrum and ().., is a nonlinear continuous mapping from DA()q ´ R into H , 0 £ q < 1 , possessing the property (), - (), £ ()r q()- Bu1 t Bu2 t M A u1 u2 (2.2) = ()q q for all u1 and u2 from the domain q DA of the operator A and such that q £ r ()r r A uj . Here M is a nondecreasing function of the parameter that does not depend on t and . is a norm in the space H . A function ut() is said to be a mild solution (in q ) to problem (2.1) on the half-interval [ss, + T) if it lies in Cs(), s+ T¢ ; q for every T ¢ < T and for all tssTÎ [ , + ) satisfies the integral equation t ()= -()ts- A + -()t - t A ()t()t , t ut e u0 òe Bu d .(2.3) s The fixed point method enables us to prove the following assertion on the local existence of mild solutions. 86 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h Theorem 2.1. a Î * q p Let u0 q . Then there exists T depending on and u0 q such that t problem (2.1) possesses a unique mild solution on the half-interval e * * r [ss, + T) . Moreover, either T = ¥ or the solution cannot be continued = + * 2 inq up to the moment tsT. Proof. º (), + ; On the space Cs, q Cs s T q we define the mapping t []() = -()ts- A + -()t - t A ()t()t , t Gu t e u0 òe Bu d . s []()Î (), + ; > ,Î Let us prove that Gu t Cs s T q for any T 0 . Assume that t1 t2 Î [], + < ss T and t1 t2 . It is evident that t2 -()t - t A -()t - t A []()= 2 1 []()+ 2 ()t()t , t Gu t2 e Gu t1 òe Bu d .(2.4)

t1 ® By virtue of (1.8) we have that if t2 t1 , then -()t - t A []()- 2 1 []() ® Gu t1 e Gu t1 q 0 . Therefore, it is sufficient to estimate the second term in (2.4). Equation (1.15) implies that

t2 -()t - t A òe 2 Bu()t()t , td £

t1 q

t2 q q q £ æö +tl × ()()t , t £ ò èø- t 1 d max Bu t2 t Î []ss, + T t1

- q ìüqq q q £ t - t 1 íý + l t - t max Bu()()t , t (2.5) 2 1 îþ1 - q 1 2 1 t Î []ss, + T (if q = 0 , then the coefficient in the braces should be taken to be equal to 1). Thus, = (), + ; ()= -()ts- A GC maps s, q Cs s T q into itself. Let v0 t e u0 . In Cs, q we consider a ball of the form ìü Uut= íý()Î C : uv- º max ut()- v ()t £ 1 . s, q 0 C 0 q îþs, q []ss, + T Let us show that for TGU small enough the operator maps into itself and is contractive. Since u £ 1 + u for uUÎ , equation (2.2) gives Cs, q 0 q Semilinear Parabolic Equations in Hilbert Space 87

max Bu()()t , t £ max B()0, t + t Î []ss, + T t Î []ss, + T

+ ()+ ()+ º (), 1 u0 q M 1 u0 q CT0 u0 q £ for all TT0 , where T0 is a fixed number. Therefore, with the help of (2.5) we find that []- £ 1- q × (),,q Gu v0 T C1 T0 u0 q . Cs, q Similarly we have 1 - q Gu[]- Gv[] £ T × C ()T , q M ()1 + u q uv- Cs, q 2 0 Cs, q , Î for uv U . Consequently, if we choose T1 such that 1- q (),,q £ 1- q (), q ()+ < T1 C1 T0 u0 q 1 andT1 C2 T0 M 1 u0 q 1 , we obtain that GUG is a contractive mapping of into itself. Therefore, possesses Ì a unique fixed point in UCs, q . Thus, we have constructed a solution on the seg- [], + + ment ss T1 . Taking sT1 as an initial moment, we can construct a solution []+ , ++ = ()+ on the segment sT1 sT1 T2 with the initial condition u0 us T1 . If we continue our reasoning, then we can construct a solution on some maximal half- interval [ss, + T*) . Moreover, it is possible that T* = ¥ . Theorem 2.1 is proved.

Î *= *()q, [ , + *) Exercise 2.1 Let u0 q and let T T u0 be such that ss T is the maximal half-interval of the existence of the mild solution ut() * to problem (2.1). Then we have either T < ¥ and lim ut()q = ® + * = ¥ , or T* = ¥ . tsT

Exercise 2.2 Using equations (1.16) and (2.5), prove that for any mild solution ut() to problem (2.1) on [ss, + T*) the estimate qa- ut()- u()t a £ C ()qa,,T t - t ,t, t Î []ss, + T , (2.6) Î ££aq £ * is valid, provided u0 q ,0 , andTT . Î () Exercise 2.3 Let u0 q and let ut be a mild solution to problem (2.1) on the half-interval [ss, + T*) . Then () Î (),,+ ÇÇ()+ d, + , ut Cs s T q Cs sT 1- s 1 Ç C ()s + d, sT+ , -s for any s > 0 , 0 <

It is frequently convenient to use the Galerkin method in the study of properties of mild solutions to the problem of the type (2.1). Let Pm be the orthoprojector in H 88 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C {},,,¼ Galerkin approximate solution h onto the span of elements e1 e2 em . Galerkin approximate solution a of the order me with respect to the basis {} is defined as a continuously diffe- p k t rentiable function e r m u ()t = g ()t e (2.7) 2 m å k k k = 1

with the values in the finite-dimensional space Pm H that satisfies the equations d ()+ ()= ()(), > = um t Aum t PmBum t t ,,ts um Pmu0 .(2.8) dt ts= It is clear that (2.8) can be rewritten as a system of ordinary differential equa- () tions for the functions gk t .

Exercise 2.4 Show that problem (2.8) is equivalent to the problem of find- () ing a continuous function um t with the values in PmH that satisfies the integral equation t ()= -()ts- A + -()t - t A ()t()t , t um t e Pm u0 òe Pm Bum d .(2.9) s

Exercise 2.5 Using the method of the proof of Theorem 2.1, prove the local solvability of problem (2.9) on a segment []ss, + T , where the parameter T > 0 can be chosen to be independent of m . Moreover, the following uniform estimate is valid: () < = ,,,¼ max um t q R ,,(2.10)m 123 []ss, + T where R > 0 is a constant.

The following assertion on the convergence of approximate functions to exact ones holds.

Theorem 2.2. Î Let u0 q . Assume that there exists a sequence of approximate solu- () [], + tions um t on a segment ss T for which estimate (2.10) is valid. Then there exists a mild solution ut() to problem (2.1) on the segment []ss, + T and ()- () £ æö()- + 1 max ut um t q C èø1 Pm u0 q , (2.11) []ss, + T l1- q m + 1 where CC= ()q,,RT is a positive constant independent of s . Semilinear Parabolic Equations in Hilbert Space 89

Proof. Let nm>q . We use (2.9), (1.14) and (1.17) to find that for > 0 we have ()- () £ ()- + un t um t q Pn Pm u0 q

t q æöq q -l ()t - t ++ + l e m + 1 Bu()t()t , t d ò èøt - t m + 1 n s t æöq q + e-q Bu()()t , t- B()tu ()t , t d . òèøt - t n m s Therefore, equations (2.2) and (2.10) give us that ()- () £ ()- + un t um t q Pn Pm u0 q

æö ++max B()0, t + MR()R J ()ts, èø[]ss, + T m t + ()qq -q ()- t -q ()t - ()t t MR e ò t un um qd , (2.12) s where t q æöq q -l ()t - t J ()ts, = + l e m + 1 dt . m ò èøt - t m + 1 s (), £ (), -¥ It is evident that Jm ts Jm t . By changing the variable in the integral xl= ()- t m + 1 t , we obtain æö¥ ()l, -¥ = -q1 + ç÷+ qq x-q -x x º l-q1 + ()+ Jm t m + 1 ç÷1 ò e d m + 1 1 k . èø0 Thus, equation (2.12) implies a ()q, R u ()t - u ()t £ ()P - P u ++1 n m q n m 0 q l1- q m + 1 t + ()q, ()- t -q ()t - ()t t a2 R ò t un um q d . s Hence, if we use Lemma 2.1 which is given below, we can find that æö1 u ()t - u ()t £ CP()- P u + (2.13) n m q èøn m 0 q l1- q m + 1 90 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C Î [], + >q h for all tssT , where C 0 is a constant depending on , RT , and . It is also a evident that estimate (2.13) remains true for q = 0 . It means that the sequence of p {}() (), + ; t approximate solutions um t is a Cauchy sequence in the space Cs s T q . e Therefore, there exists an element ut()Î Cs(), s+ T; such that equation r q (2.11) holds. Estimates (2.10) and (2.11) enable us to pass to the limit in (2.9) and 2 to obtain equation (2.3) for ut() . Theorem 2.2 is proved.

Exercise 2.6 Show that if the hypotheses of Theorem 2.2 hold, then the estimate a a ()- () £ 1 ()- + 2 max ut u t a 1 P u q - a []ss, + T m lqa- m 0 l1 m + 1 m + 1 ££aq is valid with 0 . Here a1 and a2 are constants independent of q , RT , and .

The following assertion provides a simple sufficient condition of the global solvability of problem (2.1).

Theorem 2.3. Assume that the constant M ()r in (2.2) does not depend on r, i.e. the mapping Bu(), t satisfies the global Lipschitz condition (), - (), £ - Bu1 t Bu2 t Mu1 u2 q (2.14) Î > for all uj q with some constant M 0 . Then problem (2.1) has a unique [ , +¥) Î mild solution on the half-interval s , provided u0 q . Moreover, for any two solutions u1 and u2 the estimate a ()ts- ()- () £ 2 ()- () ³ u1 t u2 t q a1 e u1 s u2 s q ,,,ts, (2.15) ql holds, where a1 and a2 are constants that depend on , 1 , and M onlyonly..

The proof of this theorem is based on the following lemma (see the book by Henry [3], Chapter 7).

Lemma 2.1. Assume that j ()t is a continuous nonnegative function on the interval ()0, T such that t -g -g j () £ 0 + ()- t 1 jt()t Î (), t c0 t c1 ò t d ,,t 0 T (2.16) 0 , ³ £ g , g < = where c0 c1 0 and 0 0 1 1 . Then there exists a constant K = ()g ,, K 1 c1 T such that c -g j () £ 0 0 ()g ,, t - g t K 1 c1 T . (2.17) 1 0 Semilinear Parabolic Equations in Hilbert Space 91

Proof of Theorem 2.3. Let ut() be a solution to problem (2.1) on the maximal half-interval of its existence [ss,)+ T . Assume that T < ¥ . Condition (2.14) gives us that (), ££(), + ()+ Bu t B 0 t Muq M0 T Muq for all u Î q and t Î []0, T . Therefore, from (2.3) and (1.15) we find that t q æöq q ut() £ u + + l ()tM ()T + Mu()t . q 0 q ò èøt - t 1 0 q d s Hence, for tssTÎ [], + we have that t () £ (),,q + (), q ()- t -q ()t t ut q C0 u0 q T C1 T ò t u qd . s

Therefore, Lemma 2.1 implies that the value ut()q is bounded on [ss,)+ T which is impossible (see Exercise 2.1). Thus, the solution exists for any half-interval [ss,)+ T. For the proof of estimate (2.15) we note that, as above, inequalities (2.3) ()= ()- () and (1.15) for the function wt u1 t u2 t give us that t q () £ () + æöq + lq ()t t wt q ws q ò èøt - t 1 Mw qd . s If we apply Lemma 2.1, we find that () £ ()ql,, () ££+ wt q C 1 M ws forsts1 . Therefore, the estimate n + 1 wt()q ££C ws()q Ctsexp {}()- ln C ws()q holds for tsn= ++s , where n is natural and 0 ££s 1 . Thus, Theorem 2.3 is proved.

Exercise 2.7 Using estimate (1.14), prove that if the hypotheses of Theo- rem 2.3 hold, then the inequality ()()- () £ QN u1 t u2 t q

ìü -l + ()ts- a3 a ()ts- £ íýe N 1 + e 2 u ()s - u ()s (2.18) îþl1- q 1 2 q N + 1 () () = - is valid for any two solutions u1 t and u2 t . Here QN IPN {}, ¼ andPN is the orthoprojector onto the span of e1 eN , the num- l q ber a2 is the same as in (2.15) and a3 depends on 1 , , and M .

Let us consider one more case in which we can guarantee the global solvability of problem (2.1). Assume that condition (2.2) holds for q = 12¤ and 92 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C ()= - ()+ (), h Bu B0 u B1 ut,(2.19) a p (), q= ¤ where B1 ut satisfies the global Lipschitz condition (2.14) with 12 and t () = e B0 u is a potential operator on the space V 12¤ . This means that there exists r () ()= ¢() a Frechét differentiable functional Fu on VB such that 0 u F u , i.e. 2 lim1 Fu()+ v - Fu()- ()B ()u , v = 0 . ® v 0 v 12¤ 0 12¤

Theorem 2.4. Let (2.2) be valid with q = 12¤ and let decomposition (2.19) take place. Assume that the functional Fu() is bounded below on V = 12¤ . Then problem (2.1) has a unique mild solution ut()Î CssT([], + ; DA())12 on an arbitrary segment []ss, + T.

Proof. () [], + Let um t be an approximate solution to problem (2.1) on a segment ss T , where Tm does not depend on (see Exercise 2.5):

d u ()t + Au ()t = P Bu()()t , t ,.(2.20)u ()s = P u dt m m m m m m 0 \ Multiplying (2.20) by u ()t = d u ()t scalarwise in the space H , we find that m dt m ìü \ 2 + d 1 12 2 + ()= ()(), , \ £ um íýA u Fu B u t um dt îþ2 m m 1 m \ £ 1 u 2 ++B ()0, t 2 M2 A12 u 2 . 2 m 1 m Since Fu() is bounded below, we obtain that

d Wu()()t £ aW() u ()t ++bB()0, t 2 dt m m 1 with constants abm and independent of , where

Wu()= 1 A12 u 2 + Fu(). 2 Therefore, Gronwall’s lemma gives us that t æö()- ()- t Wu()()t £ Wu()()s + b eat s + 2 eat B ()0, t 2 t m èøm a ò 1 d s for all tssT in the segment [], + of the existence of approximate solutions. Firstly, this estimate enables us to prove the global existence of approximate solutions (cf. Exercise 2.1). Secondly, by virtue of the continuity of the functional W on = ® ¥ V 12¤ Theorem 2.2 enables us to pass to the limit m on an arbitrary segment []ss, + T and prove the global solvability of limit problem (2.1). Theorem 2.4 is proved. Examples 93

Exercise 2.8 Let ut() be a mild solution to problem (2.1) such that () £ ££+ ut q CT for stsT . Use Lemma 2.1 to prove that () £ qa- Î [], + ut a CT t u0 q ,(2.21)tssT qa£ < = ()aq, for all 1 , where CT CT is a positive constant. Exercise 2.9 Let ut() and vt() be solutions to problem (2.1) with the ini- , Î () + () £ tial conditions u0 v0 q and such that ut q vt q CT for tssT lying in a segment [], + . Then ()- () £ ()qa, qa- - ut vt a CT t u0 v0 q , tssTÎqa[], + ,.£ < 1

Thus, if Bu(), t º Bu() and the hypotheses of Theorem 2.3 or 2.4 hold, then equa- () , tion (2.1) generates a dynamical system q St with the evolutionary operator St = () () which is defined by the equality St u0 ut , where ut is the solution to problem (2.1). The semigroup property of St follows from the assertion on the uniqueness of solution.

Exercise 2.10 Show that Theorem 2.4 holds even if we replace the assumption of semiboundedness of Fu() by the condition Fu()³ ³ -ba A12 u 2 - for some a 0 .

§3 Examples

Here we consider several examples of an application of theorems of Section 2. Our presentation is brief here and is organized in several cycles of exercises. More detailed considerations as well as other examples can be found in the books by Henry, Babin and Vishik, and Temam from the list of references to Chapter 2 (see also Sec- tions 6 and 7 of this chapter). We first remind some definitions and notations. Let W be a domain in Rd ()d ³ 1 . The Sobolev space Hm()W of the order mm ()= 012,,,¼ is defined by the formula Hm()W = {}fLÎ 2()W : D j fLÎ 2()W , jm£ , = (),,¼ , = ,,,¼ = ++¼ where jj1 j2 jd , jk 012 , jj1 jd , j j j D j f = ¶ 1 ¶ 2 ¼¶ d fx(),.xx= (),,¼ x x1 x2 xd 1 d The space Hm()W is a separable Hilbert space with the inner product 94 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h ()uv, = D j ux()D j vx()dx . a m å ò p jm£ W t e Below we also use the space H m()W which is constructed as the closure in Hm()W r 0 ¥()W of the set C0 of infinitely differentiable functions with compact support. For 2 more detailed information we refer the reader to the handbooks on the theory of So- bolev spaces.

Example 3.1 (nonlinear heat equation)

ì ¶ = nD + (),,,Ñ , Î W, > ï t u uftxu u x t 0, í (3.1) ï u ==0, u u . î ¶W t = 0 0 Here W is a bounded domain in Rd , Dn is the Laplace operator, and is a positive constant. Assume that ft(),,, x u p is a continuous function of its variables which satisfies the Lipschitz condition æöd 12 (),,, - (),,, £ - 2 + - 2 ft x u p ft x u q Kuç÷1 u2 å pj qj (3.2) èøj = 1 with an absolute constant KBut . It is clear that the operator (), defined by the formula Bu(), t()x = ft(),, x ux()Ñ, ux(), can be estimated as follows: 12 Bu(), t - Bu(), t £ Kuæö- u 2 + ¶ ()u - u 2 .(3.3) 1 2 èø1 2 å xj 1 2 j Here and below . is the norm in the space HL= 2()W . It is well-known that the operator A =¶W-D with the Dirichlet boundary condition on is a positive ()= 2()W Ç 1()W operator with discrete spectrum. Its domain is DA H H0 . More- ()12 = 1()W over, DA H0 . We also note that d ¶ 2 12 2 ==(), æöu Î ()12 = 1()W A u Au u å òèø¶ dx , uDA H0 . xj j = 1 W Therefore, equation (3.3) for Bu(), t gives us the estimate æö12 Bu(), t - Bu(), t £ K 1 + 1 A12 u , 1 2 èøl 1 l -D where 1 is the first eigenvalue of the operator with the Dirichlet boundary condition on ¶W . Therefore, we can apply Theorem 2.3 with q = 12¤ to problem (3.1). This theorem guarantees the existence and uniqueness of a mild solu- (), 1()W tion to problem (3.1) in the space C R+ H0 . Examples 95

Exercise 3.1 Assume that ft(),,, x u p º ft(),, x u is a continuous function of its arguments satisfying the global Lipschitz condition with respect to the variable u . Prove the global theorem on the existence of mild solutions to problem (3.1) in the space HL= 2()W .

Example 3.2 Let us consider problem (3.1) in the case of one spatial variable:

ì ¶ = n¶2 - (), + (),,,¶ > , Î (), ï t u x ugxu ft x u x u , t 0 x 01 í (3.4) ï u ==u 0, u =u . î x = 0 x = 1 t = 0 0 Assume that gx(), u is a continuously differentiable function with respect to ¢ (), £ () () the variable ug and u xu hu , where hu is a function bounded on every compact set of the real axis. We also assume that the function ftxup(),,, is continuous and possesses property (3.2). For any element uDAÎ=()12 = 1 (), H0 01 the following estimates hold: () £ ¢ £ ¢ max ux u 2 andu 2 u 2 , []01, L ()01, L ()01, L ()01, = ¶ where u x u . Therefore, it is easy to find that the inequality (), - (), £ ()r 12 ()- Bu1 t Bu2 t M A u1 u2 (3.5) is valid for Bu(), t = -gx(), ux()+ ft(),, x ux(), u¢()x , 12 º ¢r£ = provided A uj uj . Here . is the norm in the space H = L2()01, and M ()r = sup {}h ()x : xr< + K 2 . Equation (3.5) and Theo- 1()W rem 2.1 guarantee the local solvability of problem (3.4) in the space H0 . Moreover, if the function y ()xy, = ògx()x, x d 0 is bounded below, then we can use Theorem 2.4 to obtain the assertion on the existence of mild solutions to problem (3.4) on an arbitrary segment []0, T .

It should be noted that the reasoning in Example 3.2 is also valid for several spatial variables. However, in order to ensure the fulfilment of the estimate of the form (3.5) one should impose additional conditions on the growth of the function hu() . For example, we can require that the equation ()£ + p hu C1 C2 u be fulfilled, where p £ 2¤ ()d - 2 if d > 2 and pd is an arbitrary number if = 2 . In this case the inequality of the form (3.5) follows from the Hölder inequality and 96 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C 1()W q()W = h the continuity of the embedding of the space H into L , where q a = 2 dd¤ ()- 2 if d = dim W > 2 and qd is an arbitrary number if = 2 , q ³ 1 . p t The results shown in Examples 3.1 and 3.2 also hold for the systems of parabo- e lic equations. For example, a system of reaction-diffusion equations r ì ¶ u = nDuftu+ (),,Ñu , 2 ï t í ¶u (3.6) ==0, u u ()x , ï ¶ t = 0 0 î n ¶W W Ì d = ( ,,,¼ can be considered in a smooth bounded domain R . Here uu1 u2 ) (),,x ´ ()m + 1 d m um and ft u is a continuous function from R+ R into R such that ft(),, u x - ft(),, v h £ Mu()-xh v+ - ,(3.7) where Muv is a constant, , ÎxhRm , , Î Rmd and n is an outer normal to ¶W .

Exercise 3.2 Prove the global theorem on the existence and uniqueness of = []1()W m º 1()W ´ ¼´ mild solutions to problem (3.6) in 12¤ H H ´ H1()W .

Example 3.3 (nonlocal Burgers equation). - n + ()w, = () << > ì ut uxx u ux ft,0 xl, t 0 í (3.8) u ==u 0, u =u . î x = 0 xl= t = 0 0 Here ft() is a continuous function with the values in L2 ()0, l , l w ÎwL2()0, l , ()w, u = ò ()x ux(), tdx 0 and n is a positive parameter. Exercises 3.3–3.6 below answer the question on the solvability of problem (3.8).

Exercise 3.3 Prove the local existence of mild solutions to problem (3.8) = 1 (), in the space 12¤ H0 0 l . Hint: = -n¶2 ()= 2(), Ç 1 (), A xx,,DA H 0 l H0 0 l ()= - () w, + () Bu u ux ft.

Exercise 3.4 Consider the Galerkin approximate solutions to problem (3.8) m p k u ()t º u ()tx, = 2 × g ()t sin x . m m l å k l k = 1 Write out the system of ordinary differential equations to determine {}() gk t . Prove that this system is locally solvable. Examples 97

Exercise 3.5 Prove that the equations

1 d u ()t 2 + n¶u ()t 2 = ()ft(), u ()t (3.9) 2 dt m x m m and d ¶ u ()t 2 + n¶2 u ()t 2 £ dt x m xx m £ 2 æö()w, 2 ¶ ()2 + ()2 n èøu x um t ft (3.10) are valid for any interval of the existence of the approximate solution () 2(), um t . Here . is the norm in L 0 l . Exercise 3.6 Use equations (3.9) and (3.10) to prove the global existence of the Galerkin approximate solutions to problem (3.8) and to obtain the uniform estimate of the form ¶ () £ ()¶ , Î [], x um t C x u0 T ,(3.11)t 0 T for any T > 0 .

Thus, Theorem 2.2 guarantees the global existence and uniqueness of weak solu- = 1(), tions to problem (3.8) in 12¤ H0 0 l .

Example 3.4 (Cahn-Hilliard equation).

ì u + n¶4 u - ¶2 ()u3 ++au2 bu = 0, x Î ()0, l , t > 0, ï t x x ï í ¶ u ==¶3 u 0, ¶ u ==¶3 u 0, (3.12) ï x x = 0 x x = 0 x xl= x xl= ï u = u ()x , î t = 0 0 where n > 0 ,a , andb Î R are constants. The result of the cycle of Exercises 3.7–3.10 is a theorem on the existence and uniqueness of solutions to problem (3.12).

Exercise 3.7 Prove that the estimate ¶2 ()× 2 £ ()2 + ¶2 2 ()2 + ¶2 2 x uv Cu x u v x v is valid for any two functions ux() and vx() smooth on []0, l . Use this estimate to ascertain that problem (3.12) is locally uniquely solvable in the space ìü = Î 2(), == VuHíý0 l : ux ux 0 îþx = 0 xl= n¶4 + ® = ()12 (Hint: x 1 A ,VDA ). 98 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C () h Let us consider the Galerkin approximate solution um t to problem (3.12): a m p p k t ()º (), = 1 ()+ 2 × () um t um tx g0 t å gk t cos x . e l l l r k = 1

2 Exercise 3.8 Prove that the equality l d æö2 2 Wu()¶()t + []n¶ u ()tx, - pu()()tx, dx = 0 (3.13) dt m òèøx x m m 0 holds for any interval of the existence of approximate solutions {}() ()= 3 ++2 um t . Here pu u au b and l æön 2 a b Wu()= u +++1 u4 u3 u2 dx .(3.14) òèø2 x 4 3 2 0

In particular, equation (3.13) implies that approximate solutions exist for any seg- [], Î ment 0 T . For u0 V they can be estimated as follows: ¶ () + (), £ Î [], x um t max um tx CT ,,(3.15)t 0 T x Î []0, l

where the number CT does not depend on m .

Exercise 3.9 Using (3.15) show that the inequality

d ¶2 u ()t 2 + ¶4 u ()t 2 £ C ()1 + ¶2 u ()t 2 , t Î (0,]T dt x m x m T x m holds for any interval ()0, T and for any approximate solution u ()t to problem (3.12). (Hint: first prove that u¢ 2 £ m L4()0, l £ ()× ¢ Cuxmax u 2(), ). []0, l L 0 l Exercise 3.10 Using Theorem 2.2 and the result of the previous exercise, prove the global theorem on the existence and uniqueness of weak solutions to the Cahn-Hilliard equation (3.12) in the space V (see Exercise 3.7).

Example 3.5 (abstract form of two-dimensional system of Navier-Stokes equations) In a separable Hilbert space H we consider the evolutionary equation u d ++nAu bu(), u = ft(),,(3.16)u = u dt t = 0 0 Examples 99

where A is a positive operator with discrete spectrum, n is a positive parame- (), ()12 ´ ()12 = ter and buv is a bilinear mapping from DA DA into -12¤ = DA()-12 possessing the property ()buv(), , v = 0 for alluv, Î DA()12 (3.17) and such that for all uv, Î DA() the estimates ¤d- ¤d+ buv(), £ C A12 uA× 12 v ,(3.18)0 <

Exercise 3.11 Prove that Theorem 2.1 on the local solvability with q= =b12¤b+ is applicable to problem (3.16), where is a number from the interval ()012, ¤ .

{} < l ££l ¼ Let ek be the basis of eigenvectors of the operator A , let 0 1 2 be the corresponding eigenvalues and let Pm be the orthoprojector onto the span {},,¼ ofe1 em . We consider the Galerkin approximations of problem (3.16): ì d ()++ n () ()(), () = () ï um t Aum t Pm bum t um t Pm ft , í dt (3.20) ï ()= î um 0 Pm u0 .

Exercise 3.12 Show that the estimates t t u ()t 2 + n A12 u ()t 2 t £ u 2 + 1 f ()t 2 t (3.21) m ò m d 0 l n ò d 1 0 0 and -l n t æö2 -l n t u ()t 2 £ u ()0 2 e 1 + 1 F ()1 - e 1 (3.22) m m l èøn 1 are valid for an arbitrary interval of the existence of solutions to problem (3.20). Here FA= sup -12 ft() . Using these estimates, prove t ³ 0 the global solvability of problem (3.20).

Exercise 3.13 Show that the estimate d A12 u ()t 2 + n Au ()t 2 £ dt m m £ 2 ()()(), () 2 + ()2 n bum t um t ft (3.23) holds for a solution to problem (3.20). 100 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h Using the interpolation inequality (see Exercise 1.13) and estimate (3.18), it is easy a to find that p t 2 n2 (), 2 ££× 12 2 æöC 2 12 4 + 2 e buu Cu A u Au èøn u A u Au . r 4 2 Therefore, equation (3.23) implies that d A12 u ()t 2 £ s ()t A12 u ()t 2 + 2 ft()2 ,(3.24) dt m m m n s ()= ()2 ¤ n3 ()2 × 12 ()2 where m t 2 C um t A um t . Hence, Gronwall’s inequality gives us that

ìüt t ìüt ïï ïï 12 ()2 £ 12 2 s ()tt + 2 ()t 2 s ()xx t A um t A u0 expíým d n f expíým d d . ïïò ò ïïò îþ0 0 îþ0

It follows from equation (3.21) that the value t s ()tt d is uniformly bounded with ò0 m respect to m on an arbitrary segment []0, T . Consequently, the uniform estimates 12 () £ Î [], Î ()12 A um t CT ,,t 0 T u0 DA (3.25) are valid for any T > 0 and m = 12,,¼ .

Î Exercise 3.14 Using equations (3.23) and (3.25), prove that if u0 Î DA()12 , then n d A12 u ()t 2 + Au ()t 2 £ C , t Î []0, T dt m 2 m T for any T > 0 . <

Exercise 3.16 Use the results of Exercises 3.14 and 3.15 and inequality (3.19) to prove the global existence of mild solutions to problem ()12¤b+ Î ()12¤b+ (3.16) in the space DA , provided that u0 DA and ft() is a continuous function with the values in DA()b . Here b is an arbitrary number from the interval ()012, ¤ . Existence Conditions and Properties of Global Attractor 101

§ 4 Existence Conditions and Properties of Global Attractor

In this section we study the asymptotic properties of the dynamical system generated by the autonomous equation u d + Au = Bu(),,u = u (4.1) dt t = 0 0 where, as before, A is a positive operator with discrete spectrum and Bu() is a nonlinear mapping from q into H such that ()- ()£ ()r q()- Bu1 Bu2 M A u1 u2 (4.2) , Î = ()q q £ r £ q < for all u1 u2 q DA possessing the property A uj , 0 1 . We assume that problem (4.1) has a unique mild (in q ) solution on R+ for any Î u0 q . The theorems that guarantee the fulfilment of this requirement and also some examples are given in Sections 2 and 3 of this chapter. It should be also noted that in this section we use some results of Chapter 1 for the proof of main assertions. Further triple numeration is used in the references to the assertions and formulae of Chapter 1 (first digit is the chapter number). () , Let q St be a dynamical system with the evolutionary operator St defined = () () by the formula Stu0 ut , where ut is a mild solution to problem (4.1). () , As shown in Chapter 1, for the system q St to possess a compact global attractor, it should be dissipative. It turns out that the condition of dissipativity is not only necessary, but also sufficient in the class of systems considered.

Lemma 4.1. () , = {}q £ Let q St be dissipative and let B0 u : A uR0 be its absorbing a ball. Then the set Ba = {}u : A uR£ a is absorbing for all aqÎ (), 1 , where aq- aa R = ()aq- R + sup {}Bu(): u Î , AquR£ .(4.3) a 0 1- a q 0

Proof. Using equation (2.3) and estimate (1.17), we have t + 1 a aq- æöa Aaut()aq+ 1 £ ()- Aqut() + But()() dt , ò èøt + 1- t t ()= £ where ut Stu0 . Let B be a bounded set in q . Then St y q R0 for all ³ Î ttB and yB . Therefore, the estimate ()() £ {}() Î ³ But sup Bu : uB0 , ttB ()= a ()+ £ ³ holds for ut St y . Hence, A ut 1 Ra for all ttB . This implies the assertion of the lemma. 102 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h Theorem 4.1. a () , p Assume that the dynamical system q St generated by problem (4.1) t is dissipative. Then it is compact and possesses a connected compact global e r attractor A. This attractor is a bounded set in a for qa£ < 1 and has 2 a finite fractal dimensiondimension.. Proof. By virtue of Theorem 1.1 the space a is compactly embedded into q for aq> () , . Therefore, Lemma 4.1 implies that the dynamical system q St is com- () , pact. Hence, Theorem 1.5.1 gives us that q St possesses a connected compact a global attractor A . Evidently, A uR£ a for all u ÎA , where Ra is defined by equality (4.3). Thus, we should only establish the finite dimensionality of the attractor. Let us apply the method used in the proof of Theorem 1.9.1 and based on

Theorem 1.8.1. Let St u1 and St u2 be semitrajectories of the dynamical system () , £ ³ = , q St such that St uj q R for all t 0 , j 12 . Then equations (2.3) and (1.17) give us that t - £ - + ()qq ()- t -q - St u1 St u2 q u1 u2 q MR ò t St u1 St u2 qdt . 0 Î £ Using Lemma 2.1, we find that for all uj q such that St uj q R the following estimate holds: - £ - ££ St u1 St u2 q Cu1 u2 q for0 t 1 , where the constant C depends on q and R . Therefore, - £ - £££t t + St u1 St u2 q CSt u1 St u2 q for 0 t 1

for the considered u1 and u2 . Consequently, - £ - £ []t + 1 - St u1 St u2 q CS[]t u1 S[]t u2 q C u1 u2 q , where []t is an integer part of the number t . Thus, the estimate of the form - £ at - St u1 St u2 q Ce u1 u2 q (4.4) is valid, where the constants Ca and depend on q and R . Similarly, Lemma 1.1 gives

-l t ()- £ N + 1 - + QN St u1 St u2 q e u1 u2 q

t q æöq q -l ()t - t + MR() + l e N + 1 S u - S u dt (4.5) ò èøt - t N + 1 t 1 t 2 q 0 , Î £ = - for all u1 u2 q such that St uj q R , where QN IPN and PN is the orthoprojector onto the first NA eigenvectors of the operator . If we substitute equation (4.4) in the right-hand side of inequality (4.5), then we obtain that Existence Conditions and Properties of Global Attractor 103

-l t ()- £ ()N+1 + ()at () - QN St u1 St u2 q e CM R e JN t u1 u2 q , where t q æöq q -l ()t - t J ()t = + l e N +1 dt . N ò èøt - t N +1 0 xl= ()- t ()£ After the change of variable N +1 t it is easy to find that JN t £ l-q1 + C0 N + 1 . Therefore,

æö-l + t CR(), q Q ()S u - S u £ ç÷e N 1 + eat u - u (4.6) N t 1 t 2 q èøl1- q 1 2 q N + 1 , Î £ > for all u1 u2 q such that St uj q R . Hence, there exist t0 0 and N such that Q ()S u - S u £ d u - u ,,0 <

Equations (4.4) and (4.6) which are valid for any R > 0 enable us to prove the exis- () , tence of a fractal exponential attractor of the dynamical system q St in the same way as in Section 9 of Chapter 1.

Theorem 4.2. () , Assume that the dynamical system q St generated by problem (4.1) is dissipative. Then it possesses a fractal exponential attractor ((inertialinertial set).

Proof. It is sufficient to verify that the hypotheses of Theorem 1.9.2 (see (1.9.12)– () , (1.9.14)) hold for q St . Let us show that = = {}a £ KSÈ t Ba , Ba u : A uRa ³ tt0 can be taken for the compact K in (1.9.12)–(1.9.14). Here a is a number from the interval ()q, 1 and Ra is defined by formula (4.3). We choose the parameter = () Ì ³ t0 t0 Ba such that St KBa for tt0 . Since K is a bounded invariant set, , Î equation (4.4) is valid for any u1 u2 K with some constants Ca and . It is also easy to verify that Kk is a compact. Indeed, let {}Ì K . Then k = S y , there- n n tn n ® ® Î ® < ¥ with we can assume that kn w , yn yBa and either tn t* or ® ¥ tn . In the first case with the help of (4.4) we have 104 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C at h S y - S y £ Ce n y - y + S yS- y . a tn n t* q n q tn t* q p t Therefore, k = S y ® S yKÎ . In the second case e n tn n t* r = Î w()ÌÌ() wSlim t yn Ba St Ba K . 2 n ® ¥ n 0 Here w()Ba is the omega-limit set for the semitrajectories emanating from Ba . Thus, K is a compact invariant absorbing set. In particular this means that condition (1.9.12) is fulfilled, therewith we can take any number for g > 0 . Conditions (1.9.13) and (1.9.14) follow from equations (4.4) and (4.6). Consequently, it is sufficient to apply Theorem 1.9.2 to conclude the proof of Theorem 4.2.

() , Thus, the dissipativity of the dynamical system q St generated by problem (4.1) guarantees the existence of a finite-dimensional global attractor and an inertial set. Under some additional conditions concerning Bu() the requirement of dissipativity can be slightly weakened. We give the following definition. Let aq£ . The dynami- () , * > cal system q St is said to be a -dissipative if there exists Ra 0 such that = () for any set B bounded in a there exists t0 t0 B such that a º £ * Î Ç ³ A St ySt y a Ra for allyB q andtt0 .

Lemma 4.2. Assume that Bu() satisfies the global Lipschitz condition ()- ()£ q()- Bu1 Bu2 MA u1 u2 . (4.7) () , Let the dynamical system q St generated by mild solutions to prob- aqÎ( -], q () , lem (4.1) be a -dissipative for some 1 . Then q St is a compact dynamical system, i.e. it possesses an absorbing set which is compact in q .

Proof. () , By virtue of Lemma 4.1 it is sufficient to verify that the system q St is dissipative (i.e. q -dissipative). If we use expression (2.3) and equation (1.17), then we obtain ts+ æöqa- qa- æöq q Aqut()+ s £ Aa ut() + Bu()t()t d èøs ò èøts+ - t t ()= ()£ ()+ q for positive ts and . Here utSt u0 . Since Bu B 0 MAu , we have the estimate qa- ìüqq q æö1 qa- a A ut()+ s £ èø íý()qa- A ut() + B()0 + s îþ1- q Existence Conditions and Properties of Global Attractor 105

s q + æöq q ()t+ t M òèøs - t A ut d 0 for 0 ££s 1 . Hence, we can apply Lemma 2.1 to obtain Aqut()+ s £ C ()qa,,M ()1 + Aa ut() saq- ,.0 ££s 1 £ * ³ () Î Ç Therefore, if St u0 a Ra for tt0 B and u0 B q , then the latter inequality gives us that £ ()qa,, ()+ * ³ + () St u0 q C M 1 Ra fort 1 t0 B , () , i.e. q St is a dissipative system. Lemma 4.2 is proved.

Exercise 4.1 Show that the assertion of Lemma 4.2 holds if instead of (4.7) ()= ()+ () () we suppose that Bu B1 u B2 u , where B1 u possesses () property (4.7) and B2 u is such that {}¥() a £ * < sup B2 u : A uRa .

The following assertion contains a sufficient condition of dissipativity of the dynamical system generated by problem (4.1).

Theorem 4.3. Assume that condition (4.2) is fulfilled with q = 12¤ and Bu()= = - ¢() F u is a potential operator from 12¤ into H (the prime stands for the Frechét derivative)derivative).. Let Fu()³ - a, ()bF ¢()u , u -gFu()³ -dA12 u 2 - (4.8) Îabgd for all u 12¤ , where , , , and are real parameters, therewith b >g< 0 and 1 . Then the dynamical system is dissipative in 12¤ .

Proof. In view of Theorem 2.4 conditions (4.8) guarantee the existence of the evolutionary operator St . Let us verify the dissipativity. As in the proof of Theorem 2.4 we {}() consider the Galerkin approximations um t . It is evident that 1 d u ()t 2 ++A12 u ()t 2 ()F ¢()u , u = 0 2 dt m m m m and æö\ d 1 A12 u ()t 2 + Fu()()t + u ()t 2 = 0 . dt èø2 m m m If we add these equations and use (4.8), then we obtain that ìü d íý1 u 2 ++1 A12 u 2 Fu()++()1 - g A12 u 2 b Fu()£ d. dt îþ2 m 2 m m m m 106 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h Let a 1 2 1 2 p Vu()= u +++ A12u Fu() a. t 2 2 e r Therefore, it is clear that 2 d Vu()w()t + Vu()()t £ C dt m m with some positive constants w and Cm that do not depend on . This (cf. Exer- () , cise 1.4.1) easily implies the dissipativity of the system 12¤ St , moreover, 1 A12 S u 2 £ Vu()e-w t + C ()1 - e-w t .(4.9) 2 t 0 w Theorem 4.3 is proved.

Exercise 4.2 Show that the assertion of Theorem 4.3 holds if (4.2) is fulfilled with q = 12¤ and ()= - ¢()+ () Bu F u B0 u , () () where Fu possesses properties (4.8) and B0 u satisfies the esti- ()£ + e 12 e > mate B0 u Ce A u for 0 small enough.

Let us look at the examples of Section 3 again. We assume that the function ft(),,, x u p º fx(),, u p (4.10) in Example 3.1 possesses property (3.2) and

(),,()Ñ () () £ ()l - d 2 () + ò fx ux ux uxdx 1 òu x dx C (4.11) W W Îl1()W -D for all uH0 , where 1 is the first eigenvalue of the operator with the Dirichlet boundary condition on ¶W . Here d and C are positive constants.

Exercise 4.3 Using the Galerkin approximations of problem (3.1) and Lem- ma 4.2, prove that the dynamical system generated by equation (3.1) 1()W is dissipative in H0 under conditions (3.2), (4.10), and (4.11).

Therefore, if conditions (3.2), (4.10) and (4.11) are fulfilled, then the dynamical sys- ()1()W , 1()W tem H0 St generated by a mild (in H0 ) solution to problem (3.1) possesses both a finite-dimensional global attractor and an inertial set.

Exercise 4.4 Prove that equation (4.11) holds if d ¶ (),,Ñ = (), + u fx u u f0 xu å aj ¶ , xj j = 1 (), where aj are real constants and the function f0 xu possesses the property Existence Conditions and Properties of Global Attractor 107

(), £ ()l - d 2 + Î fx uu 1 u C , uR for any x Î W .

Exercise 4.5 Consider the dynamical system generated by problem (3.4). In addition to the hypotheses of Example 3.2 we assume that (),,,¶ º (), ft x u xu 0 and the function gx u possesses the properties y y ògx()x, x d ³ -a ,yg()b x, y -gògx()x, x d ³ - 0 0 for some positive abg , , and . Then the dynamical system genera- 1(), ted by (3.4) is dissipative in H0 01 . Exercise 4.6 Find the analogue of the result of Exercise 4.5 for the system of reaction-diffusion equations (3.6).

Exercise 4.7 Using equations (3.9) and (3.10) prove that the dynamical system generated by the nonlocal Burgers equation (3.8) with ft()º º Î 2(), 1()W fL0 l is dissipative in H0 . (), Let us consider a dynamical system VSt generated by the Cahn-Hilliard equation (3.12). We remind that ìü = Î 2(), == VuHíý0 l : ux ux 0 . îþx = 0 xl=

Exercise 4.8 Let the function Wu() be defined by equality (3.14). Show that for any positive R and a the set

ìül ïï Xa, = íýuVÎ : Wu()£ R , ux()dx £ a (4.12) R ïïò îþ0 (), is a closed invariant subset in VVS for the dynamical system t generated by problem (3.12). (), Exercise 4.9 Prove that the dynamical system Xa, R St generated by the Cahn-Hilliard equation on the set Xa, R defined by (4.12) is dissipative (Hint: cf. Exercise 3.9).

In conclusion of this section let us establish the dissipativity of the dynamical system ()()12¤b+ , DA St generated by the abstract form of the two-dimensional Navier- Stokes system (see Example 3.5) under the assumption that ft()º fDAÎ ()b . (), m We consider the dynamical system Pm HSt generated by the Galerkin approximations (see (3.20)) of problem (3.16). 108 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h Exercise 4.10 Using (3.24) prove that a p æö1 t 12 m 2 £ ç÷+ 12 m 2 t ´ e tA St u0 ç÷c0 ò A St u0 d r èø0 2 ìü1 ïï2 æöm 2 12 m ´ exp íýc St u A St u dt (4.13) ïï1 òèø0 0 îþ0 < £ for all 0 t 1 , where c0 and c1 are constants independent of m . Exercise 4.11 With the help of (3.21), (3.22) and (4.13) verify the property ()()12¤b+ , of dissipativity of the system DA St in the space DA()12¤ . Deduce its dissipativity (Hint: see Exercise 3.14–3.16).

Thus, the dynamical system generated by the two-dimensional Navier-Stokes equations possesses both a finite-dimensional compact global attractor and an inertial set.

§ 5 Systems with Lyapunov Function

In this section we consider problem (4.1) on the assumption that condition (4.2) holds with q = 12¤ and Bu() is a potential operator, i.e. there exists a functional () = ()12 ¢() Fu on 12¤ DA such that its Frechét derivative F u possesses the property Bu()= -F ¢()u ,.uDAÎ ()12 (5.1) Below we also assume that the conditions Fu()³ - a,(5.2)()bF ¢()u , u -gFu()³ -dA12 u 2 - are fulfilled for all uDAÎad()12 , where ,bÎ R , >g0 , and < 1 . On the one hand, these conditions ensure the existence and uniqueness of mild (in DA()12 ) solutions to problem (4.1) (see Theorem 2.4). On the other hand, they guarantee the existence of a finite-dimensional global attractor A for the dynamical system () , 12¤ St generated by problem (4.1) (see Theorem 4.3). Conditions (5.1) and (5.2) enable us to obtain additional information on the structure of attractor.

Theorem 5.1. Assume that conditions (5.1), (5.2), and (4.2) hold with q = 12¤ . Then A(), the global attractor of the dynamical system 12¤ St generated by Systems with Lyapunov Function 109

= () problem (4.1) is a bounded set in 1 DA and it coincides with the unstable manifold emanating from the set of fixed points of the system, i.e.

A = M+()N , (5.3) where N = {}zDAÎ (): Az= B() z (for the definition of M+()N see Secti- on 6 of Chapter 1).

The proof of the theorem is based on the following lemmata.

Lemma 5.1. ()= Assume that a semitrajectory ut St u0 possesses the property a A ut() £ Ra for all t ³ 0 , where 12¤a<<1 . Then 12 a - 12¤ A ()ut()- us() £ CR()a ts- (5.4) for all ts, ³ 0 .

Proof. For the sake of definiteness we assume that ts> ³ 0 . Equation (2.3) implies that t -()- -()- t ut()- us()= ()e tsA - I us()+ òe t A Bu()t()t d . s Since t -()- e tsA - I = -òAe-()t - t Adt , s equation (1.17) gives us that t 12 ()()- () £ ()- t a - 32¤ t a () + A ut us c0 ò t d A us s t + ()- t -12 t ()()t c1 t d max Bu . ò t ³ 0 s This implies estimate (5.4).

Lemma 5.2. > = {}£ There exists R1 0 such that the set B1 u : Au R1 is absorbing () , for the dynamical system 12¤ St .

Proof. () , By virtue of Theorem 4.3 the system 12¤ St is dissipative. Therefore, it a follows from Lemma 4.1 that Ba = {}u : A uR£ a is an absorbing set,where 110 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C ¤a<< h Ra is defined by (4.3), 12 1 . Thus, to prove the lemma it is sufficient to a a consider semitrajectories ut() possessing the property A ut() £ Ra , t ³ 0 . p t Let us present the solution ut() in the form e r t -()ts- A -()t - t A 2 ut()= e us()++òe Bu()()t -tBut()() d s

-()- + A-1 ()1 - e tsA But()() . Using Lemma 5.1 we find that t -a3 + () £ ()- -12 ++()()- t 2 t () Au t C0 ts R12¤ C1 Ra ò t d C2 R12¤ . s This implies that

Au() t + 1 £ CR()a ,,t ³ 0 a provided that A ut() £ Ra for t ³ 0 . Therefore, the assertion of Lemma 5.2 follows from Lemma 4.1.

Proof of Theorem 5.1. The boundedness of the attractor A in DA() follows from Lemma 5.2. Let us () prove (5.3). We consider the Galerkin approximation um t of solutions to problem (4.1): du ()t m + Au ()t = P Bu()()t ,.u ()0 = P u dt m m m m m 0 {},,,¼ Here Pm is the orthoprojector onto the span of e1 e2 em . Let Vu()= 1 ()Au, u + Fu(),.(5.5)u Î = DA()12 2 12¤ It is clear that \ d Vu()()t ==()Au ()t + F ¢()u ()t , u ()t - P []Au ()t - Bu()()t 2 . dt m m m m m m m This implies that t ()() - ()() = -£[]()t - ()()t 2 t Vum t Vum s ò Pm Aum Bum d s

t £ - ()()t - ()()t 2 t ò PN Aum Bum d s ³ £ for ts and for any Nm , where PN is the orthoprojector onto the span of {},,¼ e1 eN . With the help of Theorem 2.2 and due to the continuity of the func- Systems with Lyapunov Function 111 tional Vu() we can pass to the limit m ® ¥ in the latter equation. As a result, we obtain the estimate t ()() + ()()t - ()()t 2 t £ ()() ³ Vut ò PN Au Bu d Vus ,ts , (5.6) s for any solution ut() to problem (4.1) and for any N ³ 1 . If the semitrajectory ()= A ® ¥ ut St u0 lies in the attractor , then we can pass to the limit N in (5.6) and obtain the equation t Vut()() + ò Au()t - Bu()()t 2dt £ Vus()() (5.7) s for any ts³³0 and ut()Î A . Equation (5.7) implies that the functional Vu() defined by equality (5.5) is the Lyapunov function of the dynamical system () , A 12¤ St on the attractor . Therefore, Theorem 1.6.1 implies equation (5.3). Theorem 5.1 is proved.

Exercise 5.1 Using (5.6) show that any solution ut() to problem (4.1) with Î u0 12¤ possesses the property t ò Au()t 2dt < ¥ ,.t > 0 0 Prove the validity of inequality (5.7) for any solution ut() to problem (4.1).

Exercise 5.2 Using the results of Exercises 5.1 and 1.6.5 show that if the hypotheses of Theorem 5.1 hold, then a global minimal attractor A () , min of the system 12¤ St has the form A = {}Î () - ()= min wDA: Aw B w 0 .

Exercise 5.3 Prove that the assertions of Theorems 4.3 and 5.1 hold if we consider the equation u d + Au = Bu()+ h ,(5.8)u = u dt t = 0 0 instead of problem (4.1). Here hH is an arbitrary element from and Bu() possesses properties (5.1), (5.2) and (4.2) with q= 12¤ ()= ()- (), (Hint: Vh u Vu hu ).

Exercise 5.4 Let St be an evolutionary operator of problem (5.8). Show > ³ > that for any R 0 there exist numbers aR 0 and bR 0 such that a t 12 ()- £ R 12 ()- A St u1 St u2 bR e A u1 u2 , 12 £ = , ()££() provided A uj R , j 12 (Hint: Vh St uj Vh uj CR ). 112 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h Theorem 5.1 and the reasonings of Section 6 of Chapter 1 reduce the question on the a structure of global attractor to the problem of studying the properties of stationary p t points of the dynamical system under consideration. Under some additional condi- e tions on the operator Bu() it can be proved in general that the number of fixed r points is finite and all of them are hyperbolic. This enables us to use the results of 2 Section 6 of Chapter 1 to specify the attractor structure. For some reasons (they will be clear later) it is convenient to deal with the fixed points of the dynamical system generated by problem (5.8). Thus we consider the equation Lu()º Au- B() u = h ,,uDAÎ () (5.9) where as before ABu is a positive operator with discrete spectrum, () is a nonlinear mapping possessing properties (5.1), (5.2) and (4.2) with q = 12¤ , and h is an element of H .

Lemma 5.3. For any hHÎ problem (5.9) is solvable. If MH is a bounded set in , then the set L-1()M of solutions to equation (5.9) is bounded in DA() for hMÎ . If MHL is compact in , then -1()M is compact in DA() .

Proof. Let us consider a continuous functional

Wu()= 1 ()Au, u + Fu()- ()hu, (5.10) 2 = ()12 ()³ - a Î () on 12¤ DA . Since Fu for all u 12¤ , the functional Wu possesses the property

Wu()³³1 A12 u 2 - a - A-12 hA12 u 2 1 ³ A12 u 2 - a - A-12 h 2 . (5.11) 4 In particular, this means that Wu() is bounded below. Let us consider the func- () tional Wu on the subspace Pm 12¤ (Pm is the orthoprojector onto the span ,,,¼ of elements e1 e2 em , as before). By virtue of (5.11) there exists a minimum point um of this functional in Pm 12¤ which obviously satisfies the equation - ()= Aum Pm Bum Pm h .(5.12) Equation (5.11) also implies that 1 A12 u 2 ££Wu()a++A-12 h 2 4 m m £ {}() Î ++a -12 2 inf Wu: uPmH A h . Systems with Lyapunov Function 113

Hence, 12 ££()++ a -12 2 ()+ -12 2 A um F 0 A h C 1 A h with a constant Cm independent of . Thus, equations (5.12) and (4.2) give us that ££()+ ()++()12 12 Aum Bum h B 0 hMAum A um . £ £ () Therefore, if hR , then Aum CR . This estimate enables us to extract a weakly convergent (in DA() ) subsequence {}u and to pass to the limit as mk k ® ¥ in (5.12) with the help of Theorem 1.1. Thus, the solvability of equation (5.9) is proved. It is obvious that every limit (in DA() ) point u of the sequence {} um possesses the property £ £ Au CR ifhR . (5.13) This means that the complete preimage L-1()M of any bounded set MH in is bounded in DA() . Now we prove that the mapping L is proper, i.e. the pre- -1() {} image L M is compact for a compact Mu . Let n be a sequence from -1() {}() L M . Then the sequence Lun lies in the compact M and therefore there exist an element hMÎ and a subsequence {}n such that Lu()- h ® 0 k nk as k ® ¥ . By virtue of (5.13) we can also assume that {}Au is a weakly con- nk vergent sequence in DA() . If we use the equation

Aun - Bu()- h ® 0 ,,k ® ¥ k nk Theorem 1.1, and property (4.2) with q = 12¤ , then we can easily prove that the sequence {}u strongly converges in DA() to a solution u to the equation nk Lu()= h . Lemma 5.3 is proved.

Lemma 5.4. In addition to (5.1), (5.2), and (4.2) with q = 12¤ we assume that the () Î mapping B . is Frechét differentiable, i.e. for any u 12¤ there ¢() exists a linear bounded operator B u from 12¤ into H such that Bu()+ v - Bu()- B ¢()u v = oA()12 v (5.14) Î 12 £ - ¢ () for every v 12¤ , such that A v 1 . Then the operator A B u Î = ()12 is a Fredholm operator for any u 12¤ DA .

We remind that a densely defined closed linear operator GH in is said to be Fred- holm (of index zero) if (a) its image is closed; and (b)dim Ker G = dim Ker G* < ¥ .

Proof of Lemma 5.4. It is clear that the operator GABº - ¢()u has the structure 114 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C 12 12 -12 12 h GACA==- A ()IA- C A , a - p where CHCB is a bounded operator in ()= ¢()u A 12 . By virtue of Theorem t = -12 e 1.1 the operator KA C is compact. This implies the closedness of the r image of G . Moreover, it is obvious that 2 dim Ker G = dim Ker ()IK- and dim Ker G* = dim Ker ()IK- * . Therefore, the Fredholm alternative for the compact operator gives us that dim Ker G* = dim Ker G < ¥ .

Let us introduce the notion of a regular value of the operator L seen as an element hHÎ possessing the property that for every uLÎ -1 hvº {}: Lv()= h the operator L¢()u = AB- ¢()u is invertible on H . Lemmata 5.3 and 5.4 enable us to use the Sard-Smale theorem (for the statement and the proof see, e.g., the book by A. V. Babin and M. I. Vishik [1]) and state that the set R of regular values of the mapping Lu()= Au- B() u is an open everywhere dense set in H . The following assertion is valid for regular values of the operator L .

Lemma 5.5. Let hL be a regular value of the operator . Then the set of solutions to equation (5.9) is finite.

Proof. By virtue of Lemma 5.3 the set N = {}v : Lv()= h is compact. Since h Î R , the operator L¢()u = AB- ¢()u is invertible on Hu for ÎN . It is also evident that L¢()u has a domain DA() . Therefore, by virtue of the uniform boundedness principle AL¢()u -1 is a bounded operator for u ÎN . Hence, it follows from (5.14) that Av()- w £ AL¢()v -1 × L¢()v ()vw- =

==AL¢()v -1 Bv()- Bw()- B ¢()v ()vw- oA()12 ()vw-

for any vw and in N . This implies that for every v Î N there exists a vicinity that does not contain other points of the set NN . Therefore, the compact set has no condensation points. Hence, N consists of a finite number of elements. Lemma 5.5 is proved.

In order to prove the hyperbolicity (for the definition see Section 6 of Chapter 1) of fixed points we should first consider linearization of problem (5.8) at these points. Î () Assume that the hypotheses of Lemma 5.4 hold and v0 DA is a stationary solution. We consider the problem Systems with Lyapunov Function 115

u d + ()AB- ¢()v u = 0 ,.(5.15)u = u dt 0 t = 0 0 = ()12 Its solution can be regarded as a continuous function in 12¤ DA which satisfies the equation t ()= -tA + -()t - t A ¢()()tt ut e u0 òe B v0 u d . 0 Î ()12 If u0 DA , then we can apply Theorem 2.3 on the existence and uniqueness of solution. Let Tt stand for the evolutionary operator of problem (5.15). Exercise 5.5 Prove that Tt is a compact operator in every space a , 0 £ a < 1, t > 0 .

Exercise 5.6 Prove that for any r > 0 and t > 0 the set of points of the {l spectrum of the operator Tt that are lying outside the disk : lr£} is finite and the corresponding eigensubspace is finite-dimensional. ¢() Exercise 5.7 Assume that B v0 is a symmetric operator in H . Prove that the spectrum of the operator Tt is real.

The next assertion contains the conditions wherein the evolutionary operator St belongs to the class C1 + a , a > 0, on the set of stationary points.

Theorem 5.2. Assume that conditions (5.1), (5.2), and (4.2) are fulfilled with q =12¤ . () > Let Bu possess a Frechét derivative in 12¤ such that for any R 0 + a Bu()+ v - Bu()- B ¢()u v £ CA12 v 1 ,,,a > 0 , (5.16) provided that A12 vR£ , where the constant CCuR= (), depends on u and RS only. Then the evolutionary operator t of problem (5.8) has a ()=[]()¢, Frechét derivative at every stationary point v0 . Moreover, St v0 u = Tt u , where Tt is the evolutionary operator of linear problem (5.15).

Proof. It is evident that t ìü -()t - t A S []v + u - v - T u = e íýBS()t []v + u - Bv()- B ¢()v Tt u dt . t 0 0 t ò îþ0 0 0 0 Therefore, using (1.17) and (5.16) we have 116 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h t - ¤ + a a A12 y ()t £ ()2 et[]- t 12{CA12 ()S []v + u - v 1 + p ò t 0 0 t e 0 r ü + B¢()v × A12 yt()ý dt , 2 0 þ y ()= []+ - - where t St v0 u v0 Tt u . Using the result of Exercise 5.4 we obtain that a t 12 ()[]+ - £ R 12 12 £ A St v0 u v0 bR e A u ifA uR . Thus, t ()1 + a a t + a 12 y() £ R 12 1 + ()- t -12¤ 12 yt() t A t C0 te A u C1 ò t A d . 0 Consequently, Lemma 2.1 gives us the estimate 12 ()[]+ - - £ 12 1 + a Î [], A St v0 u v0 Tt u CT A u ,.t 0 T This implies the assertion of Theorem 5.2.

Exercise 5.8 Assume that the constant CR in (5.16) depends on only, 12 £ 12 £ Î 1 + a provided that A uR and A vR . Prove that St C for any 0 <

The reasoning above leads to the following result on the properties of the set of fixed points of problem (5.8).

Theorem 5.3. Assume that conditions (5.1), (5.2) and (4.2) are fulfilled with q = 12¤ () and the operator Bu possesses a Frechét derivative in 12¤ such that equation (5.16) holds with the constant CCR= () depending only on R for A12 uR£ and A12 vR£ . Then there exists an open dense (in H ) set R Î R () , such that for h the set of fixed points of the system 12¤ St generated by problem (5.8) is finite. If in addition we assume that B ¢()z is a symmetric operator for zDAÎ () , then fixed points are hyperbolichyperbolic..

In particular, this theorem means that if h Î R , then the global attractor of the dynamical system generated by equation (5.8) possesses the properties given in Exer- cises 1.6.9–1.6.12. Moreover, it is possible to apply Theorem 1.6.3 as well as the other results related to finiteness and hyperbolicity of the set of fixed points (see, e.g., the book by A. V. Babin and M. I. Vishik [1]). Systems with Lyapunov Function 117

Example 5.1 Let us consider a dynamical system generated by the nonlinear heat equation

ì ¶u ¶2 u ï - n + gx(), ux()= hx(),, 0 < 0, í ¶t ¶x2 (5.17) ï u ==u 0, u =u ()x , î x = 0 x = 1 t = 0 0 1(), (), in H0 01 . Assume that gx u is twice continuously differentiable with respect to its variables and the conditions y y ògx()x, x d ³ -a , yg()b x, y -gògx()x, x d ³ - 0 0 are fulfilled with some positive constants ab,g , and .

Exercise 5.9 Prove that the dynamical system generated by equation (5.17) possesses a global attractor A = M+ ()N , where N is the set of stationary solutions to problem (5.17).

Exercise 5.10 Prove that there exists a dense open set R in L2()01, such that for every hx()Î R the set N of fixed points of the dynamical system generated by problem (5.17) is finite and all the points are hyperbolic.

It should be noted that if a property of a dynamical system holds for the parameters from an open and dense set in the corresponding space, then it is frequently said that this property is a generic property. However, it should be kept in mind that the generic property is not the one that holds almost always. For example one can build an open and dense set R in []01, , the Lebesgue measure of which is arbitrarily small ()£ e . To do that we should take ¥ ìü R = Î (), -e< × - k - 2 È íýx 01: xrk 2 , k = 1îþ {} [], where rk is a sequence of all the rational numbers of the segment 01 . There- fore, it should be remembered that generic properties are quite frequently encoun- tered and stay stable during small perturbations of the properties of a system. 118 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h § 6 Explicitly Solvable Model a p of Nonlinear Diffusion t e r 2 In this section we study the asymptotic properties of solutions to the following nonlinear diffusion equation ì æö1 ç÷ ï u - n u ++ ux(), t2 dx - G u r u = 0, 0 < 0, ï t xx ç÷ò x í èø0 (6.1) ï ï u ==u 0, u =u ()x , î x = 0 x = 1 t = 0 0 where n >0 , >G0 , and r are parameters. The main feature of this problem is that the asymptotic behaviour of its solutions can be completely described with the help of elementary functions. We do not know whether problem (6.1) is related to any real physical process.

Exercise 6.1 Show that Theorem 2.4 which guarantees the global existence and uniqueness of mild solutions is applicable to problem (6.1) in the 1 (), Sobolev space H0 01 . Exercise 6.2 Write out the system of ordinary differential equations for the {}() functions gk t that determine the Galerkin approximations m ()= () p um t 2 å gk t sin kx (6.2) k = 1 of the order m of a solution to problem (6.1). () Exercise 6.3 Using the properties of the functions um t defined by equation (6.2) prove that the mild solution ut(), x possesses the properties t 1 ut()2 + ()tn¶u()t 2 + u()t 4 - G u()t 2 d = 1 u 2 (6.3) 2 ò x 2 0 0 and

- np2 G2 - np2 ut()2 £ u 2 e 2 t + ()1 - e 2 t .(6.4) 0 4 np2 Here and below . is a norm in L2()01, .

Exercise 6.4 Using equations (6.3) and (6.4) prove that the dynamical sys- 1(), tem generated by problem (6.1) in H0 01 is dissipative. Explicitly Solvable Model of Nonlinear Diffusion 119

()1 (), , Therefore, by virtue of Theorem 4.1 the dynamical system H0 01 St generated by equation (6.1) possesses a finite-dimensional global attractor.

()= (), Î (); 1 (), Let ut ux t C R+ H0 01 be a mild solution to problem (6.1) with ()Î 1 (), (), the initial condition u0 x H0 01 . Then the function ux t can be considered as a mild solution to the linear problem - n ++() r = <<, > ì ut uxx btu ux 0, 0 x 1 t 0 , í (6.5) u ==u 0, u =u ()x , î x = 0 x = 1 t = 0 0 where bt() is a scalar continuous function defined by the formula 1 bt()= ò ux(), t 2dx - G . 0 We consider the function

ìüt ïïr2 r vx(), t = ux(), texp íýb()t d t+ t - x .(6.6) ïïò 4 n 2 n îþ0 Then it is easy to check that vx(), t is a mild solution to the heat equation

ì = n Î (), > ïvt vxx, x 01, t 0 , ï í (6.7) ï ìür v ==v 0, v =u ()x exp íý- x . ï x = 0 x = 1 t = 0 0 n î îþ2

The following assertion shows that the asymptotic properties of equation (6.7) completely determine the dynamics of the system generated by problem (6.1).

Lemma 6.1. 1(), (), Every mild (in H0 01 ) solution ux t to problem (6.1) can be rewritten in the form wx(), t ux(), t = , (6.8) 12 ìüt ïï íý12+ w()t 2 dt ïïò îþ0 where wx(), t has the form ìür2 r æö wx(), t = vx(), t exp íýèø G- t + x (6.9) îþ4 n 2 n and vx(), t is the solution to problem (6.7). 120 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h Proof. a Let wx(), t have the form (6.9). Then we obtain from (6.6) that p t e ìüt r ïï wx(), t = ux(), texp íý ò u()t 2dt .(6.10) 2 ïï îþ0 Therefore,

ìüt ìüt 2 2 ïï2 d ïï2 wx(), t ==ux(), t exp íý2 u()t dt 1 exp íý2 u()t dt . ïïò 2 dt ïïò îþ0 îþ0 Hence,

ìüt t ïï expíý2 u()t 2dt = 1+ 2 w()t 2dt . ïïò ò îþ0 0 This and equation (6.10) imply (6.8). Lemma 6.1 is proved.

Now let us find the fixed points of problem (6.1). They satisfy the equation

-n u ++() u 2 - G u r u ===0, u u 0 . xx x x = 0 x = 1 ìür Therefore, ux()= wx()expíý x , where wx() is the solution to the problem îþ2 n æör2 -n w + u 2 - G + w ===0, w w 0 . xx èø4 n x = 0 x = 1 However, this problem has a nontrivial solution wx() if and only if r2 wx()=C sinp nx andu 2 - G ++ np()n 2 = 0 , 4 n ìür where nuw is a natural number. Since = expíý x , we obtain the equation îþ2 n 1 r x r2 C2 en sin2 p nxdx ++np()n 2 - G = 0 ò 4 n 0 which can be used to find the constant C . After the integration we have

2 n2 p2 r2 2 n rn 2 n 2 C r ()e - 1 = G -np- ()n . r2 + 4 n2 n2 p2 4 n The constant Cn can be found only when the parameter possesses the property Gr-np2¤ ()4 n - ()n 2 > 0 . Thus, we have the following assertion. Explicitly Solvable Model of Nonlinear Diffusion 121

Lemma 6.2. r2 ggGnr=gnp()G,, º - £ 2 Let 4 n . If , then problem (6.1) has a unique ()º ()p 2 < gp£ 2 ()+ 2 ³ fixed point u0 x 0 . If n n 1 for some n 1 , then the fixed points of problem (6.1) are r x ()º ()= ± m 2n p = ,,¼ , u0 x 0 ,,,u±k x k e sin kx k 12 n (6.11) where

1 2 rd ()Grn,, m º m ()Grn,, = k , (6.12) k k n2 p r 4 k æöæö - èøexpèøn 1

d ()Grn,, = []rGn- r2 - ()p n 2 × []2 + ()p n 2 k 4 4 n 4 n .

Exercise 6.5 Show that every subspace r ìü x H = Lin íýe2 n sin p kx: k = 12,,¼ ,N (6.13) N îþ ()1(), , is positively invariant for the dynamical system H0 01 St generated by problem (6.1).

Theorem 6.1. r2 Let gGº - < np2 . Then for any mild solution ut() to problem (6.1) 4 n the estimate ¶ () £ ()rn, -()np2 - g t ¶ ³ x ut C e x u0 ,,,t 0 , (6.14) gnp³np2 2 2 > g is valid. If and N , then the subspace HN - 1 defined by formula (6.13) is exponentially attracting: (), £ ()rn, -()np2 N2 - g t ¶ dist 1 (), St u0 HN - 1 C e x u0 . (6.15) H0 01 1(), Here St is the evolutionary operator in H0 01 corresponding to (6.1).

Proof. Let wx(), t be of the form (6.9). Then r ¥ - x ()º 2n ()= -()np()k 2 - g t () wr t e wt å e Ck, r ek x ,(6.16) k = 1 ()= p where ek x 2 sin kx and 1 r - x = 2n () () Ck, r òe u0 x ek x dx . 0 122 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C gnp< 2 h Assume that . Then it is obvious that a ¥ p ¶ ()2 = ()p 2 2 {}- ()np()2 - g £ t xwr t å k Ck, r exp 2 k t e k = 1 r r æö- x 2 2 £ exp {}¶-2 ()np2 - g t ç÷e 2n u . x èø0 (6.17)

However, equation (6.8) implies that r æö- x ¶ ç÷e 2n ut() £ ¶ w ()t . x èøx r Therefore, estimate (6.14) follows from (6.17) and from the obvious equality £ ¤¶p gnp³ 2 > ()gn¤ p ¤ () u 1 x u . When and N 1 the function wt in (6.9) can be rewritten in the form r x ()= ()+ 2n () wt hN t e wr, N t ,(6.18) where r N -1 x () = 2n -()np()k 2 - g t p Î hN t 2 e å e Ck, r sin kx HN -1 k = 1 () and wr, N t can be estimated (cf. (6.17)) as follows: r 2 æö- x ¶ w ()t 2 £ exp {}¶-2 ()np()N 2 - g t ç÷e 2n u .(6.19) x r, N x èø0

Thus, it is clear that dist ()S u , H £ C ()¶rn, w ()t . 1 (), t 0 N -1 x r, N H0 01 Consequently, estimate (6.15) is valid. Theorem 6.1 is proved.

In particular, Theorem 6.1 means that if gGrº -np2 ¤ ()4 n < 2 , then the global attractor of problem (6.1) consists of a single zero element, whereas if gnp³ 2 , the attractor lies in an exponentially attracting invariant subspace H , where N = N0 0 = []()gn1¤ p ¤ and []. is a sign of the integer part of a number. The following assertion shows that the global minimal attractor of problem (6.1) consists of fixed points of the system. It also provides a description of the corresponding basins of attraction.

Theorem 6.2. r2 Let gGºp-np > 2 . Assume that the number -1 gn¤ is not integer. 4n np()2 < g Suppose that N0 is the greatest integer such that N0 . Let Explicitly Solvable Model of Nonlinear Diffusion 123

1 r - x ()= 2n () p Cj u0 2 òe u0 x sin jxdx 0 and let ut()be a mild solution to problem (6.1). ()= = ,,¼ , (a) If Cj u0 0 for all j 12 N0 , then ¶ () £ ()nr, {}¶-() np2 ()+ 2 - g ³ x ut C exp N0 1 t x u0 ,,.t 0 . (6.20) ()¹ (b) If Cj u0 0 for some j between 1 and N0 , then there exist positive =bbgn()nrg,,; = (), numbers CC u0 and such that ¶ ()()- £ -b t ³ x ut us k Ce ,,,t 0 , (6.21) () s = () where u±k x is defined by formula (6.11), sign Ck u0 , and k is ()¹ the smallest index between 1 and N0 such that Ck u0 0 .

Proof. () In order to prove assertion (a) it is sufficient to note that the value hN t is = + identically equal to zero in decomposition (6.18) when NN0 1 . Therefore, (6.20) follows from (6.19). Now we prove assertion (b). In this case equation (6.18) can be rewritten in the form

wt()= g ()t + h ()t + wr, + ()t ,(6.22) k k N0 1 where r x 2 ()= 2n -[]np()k - g t () p gk t 2 e e Ck u0 sin kx,

r N0 x 2 ()= 2n -[]np()j - g t p hk t 2 e å e Cj, r sin jx, jk= + 1 ()º = and hk t 0 if kN0 . Moreover, the estimate -()np2 ()+ 2 - g N0 1 t ¶ wr, + ()t £ C ()rn, e ¶ u (6.23) x N0 1 x 0 is valid for wr, + ()t . It is also evident that N0 1 2 ¶ () £ ()rn, ()gn-p2 ()k + 1 t ¶ x hk t C e x u0 .(6.24) Since ()2 - ()2 ££()() + () () - () wt gk t wt gk t wt gk t

£ ()2 g ()t ++h ()t wr, + ()t ()h ()t + wr, + ()t , k k N0 1 k N0 1 using (6.23) and (6.24) we obtain ()2 - ()2 £ {}¶[]gnp- 2 ()2 + ()+ 2 2 wt gk t C exp 2 k k 1 t xu0 . 124 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h Integration gives a t 1 t p r x ()tgnp- ()2 t 2 g ()t 2dt ==4 []C ()u 2 en sin2p kxdxe2 k dt e ò k k 0 ò ò r 0 0 0 C ()u 2 2 2 = æök 0 ()2 ()gnp- ()k t - 2 èøm e 1 , k m where k is defined by formula (6.12). Hence, t C ()u 2 2 + ()t 2 t = + æök 0 2 ()gnp- ()k t + () 12 w d 12èøm e ak t ,(6.25) ò k 0 where ìü2 + ()+ 2 () £ +¶æögnp- 2 k k 1 2 £ - ak t C 12exp íýèøt x u0 ,.kN0 1 îþ2 Let g ()t ()= k vk t . æöt 12 ç÷+ ()t 2 t ç÷12ò w d èø0 ££ - We consider the case when 1 kN0 1 . Equations (6.23)–(6.25) imply that ¶ h ()t ¶ ()¶()- () ££() + x k x ut vk t x wr, N +1 t 0 æöt 12 ç÷+ ()t 2 t ç÷12ò w d èø0 ì -()np2 ()N +1 2 - g t £ C ()¶rn, u íe 0 + x 0 î

exp {}()gn-p2 ()k +1 2 t ü + ý . C ()u 2 2 12 þ æö+ æök 0 2 ()gnp- ()k t + () èø12èøm e ak t k Therefore, ¶ ()()- () £ x ut vk t

2 ìü-()np2 ()N +1 - g t -np2 ()()+ 2 - 2 - £ C ()rn,,u íýe 0 + e k 1 k t Gt() 12 , 0 îþ Explicitly Solvable Model of Nonlinear Diffusion 125 where C ()u 2 ()= ()+ () -2 ()gnp- 2 k2 t + æök 0 Gt 1 ak t e 2 èøm . k It follows from (6.25) that Gt()> 0 for all t ³ 0 and G()0 = 1 . Moreover, the () above-mentioned estimate for ak t enables us to state that C ()u 2 ()= æök 0 > limGt 2 èøm 0 . t ® ¥ k = ()g,,,n > This implies that there exists a constant Gmin Gmin k u0 0 such that ()³ ³ Gt Gmin for all t 0 . Consequently, ¶ ()()- () £ ()rng,,; -a t x ut vk t C u0 e ,(6.26) aa==(),,ng {} np2 ()+ 2 -npg, 2 ()+ where k min N0 1 2 k 1 . Now we consider () the value vk t . Evidently, ()= j (), () vk t k tu0 us k x , where (){}() gn-p2 2 2 Ck u0 exp k t j ()tu, = . k 0 C ()u 2 12 m æö+ æök 0 {}()gn-p2 2 + () k èø12èøm exp 2 k t ak t k Simple calculations give us that j (), - £ ()rng,,, -b t k tu0 1 C u0 e with the constant bnp= 2 ()2 k + 1 . This and equation (6.26) imply (6.21), provi- £ - = ded kN0 1 . We offer the reader to analyse the case when kN0 on his/her own. Theorem 6.2 is proved.

Theorem 6.2 enables us to obtain a complete description of the basins of attraction of ()1(), , each fixed point of the dynamical system H0 01 St generated by problem (6.1). Corollary 6.1. Let gGrºp-np2 ¤ ()4 n > 2 . Assume that the number -1 gn¤ is not in- np()2 < g teger. Let N0 be the greatest integer possessing the property N0 . We denote 1 r - x ()= 2 n () p Cj u 2 òe uxsin jxdx 0 and define the sets D = {}Î 1 (), ()= = ,,¼ - ()> k uH0 01: Cj u 0 , j 1 k 1 , Ck u 0 , D = {}Î 1 (), ()= = ,,¼ - ()< -k uH0 01: Cj u 0 , j 1 k 1 , Ck u 0 126 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C = ,,¼ , h for k 12 N0 . We also assume that a 1 p D = {}Î (), ()= = ,,¼ , 0 uH0 01: Cj u 0 , j 12 N0 . t e Then for any l = 01,,,,± ±¼2 ±N we have that r 0 limS vu- = 0 Û v Î D , 2 t l 1(), l t ® ¥ H0 01 {} where ul are the fixed points of problem (6.1) which are defined by equalities (6.11).

The next assertion gives us a complete description of the global attractor of problem (6.1).

Theorem 6.3.

Assume that the hypotheses of Theorem 6.2 hold and N0 is the same as in Theorem 6.2. Then the global attractor A of the dynamical system ()1(), , H0 01 St generated by equation (6.1) is the closure of the set r ìüN x ïï2 0 x e2n sinp kx ïïåk = 1 k = vx()= : x Î R , (6.27) íýa 12 j ïïæöN0 kj 12+ xx ïïèøå , = k j n + n îþkj 1 k j n = gnp- ()2 = ,,¼ , where k k , k 12 N0 , and 1 r x a = n p × p , = ,,¼ , kj 2 òe sin kx sin jxdx ,,.kj 12 N0 . 0 Every complete trajectory {}ut(): t Î R which lies in the attractor and = , ± ,,¼ ± does not coincide with any of the fixed points uk , k 01 N0 , has the form r N n t x 2 0 x e k e 2n sinp kx åk = 1 k ut()= , (6.28) 12 , N a ()n + n t æö+ x0 x kj k j èø12å k j n + n e kj, = 1 k j x = ,,¼ , Î where k are real numbers, k 12 N0 , t R .

N Exercise 6.6 Show that for all x Î R 0 the function

N a ()n + n t (), x = x0 x kj k j ³ at 2 å , = k j n + n e ,(6.29)t 0 kj 1 k j is nonnegative and it is monotonely nondecreasing with respect to t ¢(), x ³ (), x ® ® -¥ (Hint: at t 0 and at 0 as t ). Explicitly Solvable Model of Nonlinear Diffusion 127

Proof of Theorem 6.3. Let h belong to the set given by formula (6.27). Then by virtue of Lemma 6.1 we have wx(), t S h = ,(6.30) t æöt 12 ç÷+ ()t 2 t ç÷12ò w d èø0 where

r N0 2 x n t wt()= e 2n x e k sinp kx 1 + a ()0, x å k k = 1 and the value at(), x is defined according to (6.29). Simple calculations show that t at(), x - a()0, x 2 w()t 2dt = . ò 1 + a()0, x 0 = () ³ () Therefore, it is easy to see that St hut for t 0 , where ut has the form (6.28). = () It follows that St and that a complete trajectory ut lying in has the form (6.28). In particular, this means that Ì A . To prove that = A it is sufficient to verify using Theorem 6.1 and the reduction principle (see Theorem 1.7.4) that for any element hHÎ there exists a semitrajectory vt()Ì such that N0 ¶ ()- () = limx St hvt 0 t ® ¥ uniformly with respect to hH from any bounded set in . To do this, it should be N0 kept in mind that for N 0 r x h = 2 x e 2n sinp kx Î H (6.31) å k N0 k = 1

St h has the form (6.30) with

r N0 x n t ()= 2n x k p wt 2 e å k e sin kx. k = 1 Therefore, it is easy to find that wt() S h = , t ()1 + at(), x - a()0, x 12 where at(), x is given by formula (6.29). Hence, if we choose wt() vt()= Î , ()1 + at(), x 12 we find that 128 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C -y() = (), h St hvt thSt h ,(6.32) a p where t e æöa()0, x 12 r y ()th, = 11- - . èø1 + at(), x 2 Using the obvious inequality 11- ()- x 12 £ x ,,0 ££x 1 we obtain that a()0, x y ()th, £ , 1 + at(), x provided that h has the form (6.31). It is evident that

t N0 2 n t (), x ³ (), x + N0 t × x2 at a 0 c0 òe d å k . 0 k = 1 Therefore, æöt -1 2 n t y (), £ ç÷N0 t th Ceç÷ò d . èø0 1(), Consequently, the dissipativity property of St in H0 01 and equations (6.32) give us that æöt -1 2 n t ¶ ()- () £ ç÷N0 t ³ x St hvt Ceç÷ò d , ttB èø0 for all hBHÎ Ì , where B is an arbitrary bounded set. Thus, = A and there- N0 fore Theorem 6.3 is proved.

Exercise 6.7 Show that the set coincides with the unstable manifold M+()0 emanating from zero, provided that the hypotheses of Theo- rem 6.3 hold.

Exercise 6.8 Show that the set = M+()0 from Theorem 6.3 can be described as follows:

ì N0 r ï x = = h 2n p ív 2 å k e sin kx : ï = î k 1 N ü æö0 a kj N ï ç÷2 hh < 1, h Î R 0 ý . èøå k j n + n , = k j ï kj 1 þ Explicitly Solvable Model of Nonlinear Diffusion 129

Therewith, the global attractor A has the form

ìüN r N0 ïï0 x a A = = h 2n p hh kj £ íýv 2 å k e sin kx:2 å k j n + n 1 . k j ïï= , = îþk 1 kj 1

Exercise 6.9 Prove that the boundary

ì N r ï 0 x ¶ = = h 2n p ív 2 å k e sin kx: ï î k = 1 ü N0 æöa N ï ç÷hh kj = h Î 0 2 å k j n + n 1, R ý èøk j ï kj, = 1 þ of the set is a strictly invariant set.

Exercise 6.10 Show that any trajectory g¶ lying in has the form {ut(), t Î}R , where N r 0 n t x h k 2n p å k e e sin kx = N ut()= k 1 ,.h Î R 0 N0 12 æöa ()n + n t hç÷h kj k j å k j n + n e èøk j kj, = 1

Exercise 6.11 Using the result of Exercise 6.10 find the unstable manifold () = ± , ± ,,¼ ± M+ uk emanating from the fixed point uk , k 12 N0 . {}, ,= Exercise 6.12 Find out for which pairs of fixed points uk uj , kj = ± , ± ,,¼ ± 12 N0 , there exists a heteroclinic trajectory connec- {}() Î ting them, i.e. a complete trajectory ukj t : t R such that = () = () uk lim ukj t ,.uj lim ukj t t ® -¥ t ® +¥

Exercise 6.13 Display graphically the global attractor A on the plane gene- = []r¤ ()2 n x p = []r¤ ()2 n x ´ rated by the vectors e1 e sin x and e2 e ´ p = sin2 x for N0 2 . 130 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h Exercise 6.14 Study the structure of the global attractor of the dynamical a p system generated by the equation t e ì r ï æö1 ç÷ ï u -n u + ()ux(), t 2 + vx(), t 2 x - G u - a v = 0, 0 < 0 , 2 ï t xx ç÷ò d ï èø ï 0 í æö1 ï -n ++ç÷()(), 2 + (), 2 - G a << > ï vt vxx ç÷ò ux t vx t dx v u = 0, 0 x 1 , t 0 , ï èø ï 0 ï u = u = v = v =0 î x = 0 x = 1 x = 0 x = 1

1(), ´1 (), nGa,,, in H0 01 H0 01 , where are positive parameters.

§ 7 Simplified Model of Appearance of Turbulence in Fluid

In 1948 German mathematician E. Hopf suggested (see the references in [3]) to consider the following system of equations in order to illustrate one of the possible sce- narios of the turbulence appearance in fluids: = m - - - ì ut uxx v * vw* w u * 1 , (7.1) ï = m + ++ í vt vxx v * uv* aw* b , (7.2) ï = m + - + î wt wxx w * uv* b w * a , (7.3)

where the unknown functions uv , , and w are even and 2 p -periodic with respect to x and 2 p 1 ()f g ()x = fx()- ygy()dy . * 2p ò 0 Here ax() and bx() are even 2 pm -periodic functions and is a positive constant. We also set the initial conditions u = u ()x , v = v ()x , w = w ()x .(7.4) t = 0 0 t = 0 0 t = 0 0 As in the previous section the asymptotic behaviour of solutions to problem (7.1)–(7.4) can be explicitly described. Let us introduce the necessary functional spaces. Let = {}Î 2 () ()= ()- = ()+ p HfLloc R : fx fx fx 2 . Simplified Model of Appearance of Turbulence in Fluid 131

Evidently H is a separable Hilbert space with the inner product and the norm defined by the formulae: 2 p ()fg, = ò fx()gx()dx , f 2 = ()ff, 0 There is a natural orthonormal basis ìü íý1 ,,1 cosx 1 cos2 x ,¼ îþ2 p p p () Î in this space. The coefficients Cn f of decomposition of the function fH with respect to this basis have the form 2 p 2 p 1 1 C ()f = fx()dx ,.C ()f = fx()cosnxd x 0 2p ò n p ò 0 0

Exercise 7.1 Let fg, Î H . Then f * gHÎ and 1 f * g £ fg× .(7.5) 2p

The Fourier coefficients Cn of the functions f * gf , , and g obey the equations 1 1 C ()f * g = C ()f C ()g ,.(7.6)C ()f * g = C ()f C ()g 0 2p 0 0 n 2 p n n

Exercise 7.2 Let pm be the orthoprojector onto the span of elements {}coskx, k = 01,,¼ ,m in H . Show that ()==() () pm f * g pm f * gf* pm g .(7.7)

Let us consider the Hilbert space H = H3 º HHH´ ´ with the norm ()uvw; ; = H = ()u 2 ++v 2 w 2 12 as the phase space of problem (7.1)–(7.4). We define an operator A by the formula (),, = ()- m ; - m ; - m (); ; Î () A uvw u uxx v vxx w wxx , uvw DA on the domain ()= []2 ()3 Ç DA Hloc R H , 2 () where Hloc R is the second order Sobolev space.

Exercise 7.3 Prove that A is a positive operator with discrete spectrum. Its ¥ eigenvalues {}l have the form: n n = 0 l ===l l + m 2 = ,,,¼ 3 k 3k + 1 3k + 2 1 k ,,k 012 132 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h while the corresponding eigenelements are defined by the formulae a ü p = 1 (); ; = 1 (); ; = 1 (); ; t e0 100, e1 010, e2 001, ï e 2p 2p 2p ï r ï e = 1 ()coskx; 00; , e = 1 ()0 ; coskx; 0 , ý (7.8) 2 3 k p 3 k +1 p ï ï e = 1 ()00; ; coskx , ï 3 k + 2 p þ

where k = 12,,¼

Let (),, = - - - b1 uvw v * vw* w u * 1, (),, = ++ b2 uvw v * uv* aw* b , (),, = - + b3 uvw w * uv* b w * a . (),, Î Equation (7.5) implies that bj uvw H , provided abuv , , , , and w are the elements of the space Hj , = 123,, . Therefore, the formula (),, = ()(),, + ; (),, + ; (),, + Bu v w b1 uvw ub2 uvw v b3 uvw w gives a continuous mapping of the space H into itself.

Exercise 7.4 Prove that æö By()- By() £ C 1 +++aby +y y - y , 1 2 H èø1 H 2 H 1 2 H = (); ; Î = , where yj uj vj wj H , j 12 .

Thus, if ab, Î H , then problem (7.1)–(7.4) can be rewritten in the form dy + Ay = By(),,y = y dt t = 0 0 where AB and satisfy the hypotheses of Theorem 2.1. Therefore, the Cauchy problem (7.1)–(7.4) has a unique mild solution yt()= ()ut(), vt(), wt() in the space H on a segment []0, T , provided that ab, Î H . In order to prove the global existence theorem we consider the Galerkin approximations of problem (7.1)–(7.4). () The Galerkin approximate solution ym t of the order 3 m with respect to basis (7.8) can be presented in the form m - 1 ()==æö()m (); ()m (); ()m () æö(); (); () ym t èøu t v t w t å èøuk t vk t wk t coskx,(7.9) k = 0 () () () where uk t , vk t , and wk t are scalar functions. By virtue of equations (7.7) it is easy to check that the functions u()m ()t , v()m ()t , and w()m ()t satisfy equations (7.1)–(7.3) and the initial conditions ()m ()= ()m ()= ()m ()= u 0 pm u0 ,,v 0 pmv0 w 0 pmw0 .(7.10) Simplified Model of Appearance of Turbulence in Fluid 133

Thus, approximate solutions exist, locally at least. However, if we use (7.1)–(7.3) we can easily find that ìü 1 × d íýu()m ()t 2 ++v()m ()t 2 w()m ()t 2 + 2 dt îþ

ìü() () () + m íýu m 2 ++v m 2 w m 2 = îþx x x

() () () () () () = -()u m * 1, u m ++()v m * av, m ()w m * aw, m . Therefore, inequality (7.5) leads to the relation

d y ()t 2 £ C ()1 + a y ()t 2 . dt m H m H () This implies the global existence of approximate solutions ym t (see Exercise 2.1). Therefore, Theorem 2.2 guarantees the existence of a mild solution to problem (7.1)–(7.4) in the space H = H3 on the time interval of any length T . Moreover, the mild solution yt()= ()ut(); vt(); wt() possesses the property ()- () ® ® ¥ maxyt ym t 0 , m []0, T [], () for any segment 0 T . Approximate solution ym t has the structure (7.9). {}(); (); () Exercise 7.5 Show that the scalar functions uk t vk t wk t involved in (7.9) are solutions to the system of equations:

ì \ + m 2 = - 2 - 2 - d ï uk k uk vk wk u0 k 0 , (7.11) ï \ 2 í vk + m k v = v u ++v a w b , (7.12) ï k k k k k k k \ ï w + m k2 w = w u -v b + w a . (7.13) î k k k k k k k k =d,,¼ = ¹d= = Here k 12 , k0 0 for k 0 , 00 1 , the numbers ak = () = () Ck a and bk Ck b are the Fourier coefficients of the functions ax() and bx() .

(), Thus, equations (7.1)–(7.3) generate a dynamical system H St with the evolutionary operator St defined by the formula = ()(); (); () St y0 ut vt wt , where ()ut(); vt(); wt() is a mild solution (in H ) to the Cauchy problem (7.1)– = (),, (7.4), y0 u0 v0 w0 . An interesting property of this system is given in the following exercise.

L {},, Exercise 7.6 Let k be the span of elements e3 k e3 k +1 e3 k +2 , where = ,,,¼ {} k 012 and en are defined by equations (7.8). Then the L subspace k of the phase space H is positively invariant with re- ()L Ì L spect to St St k k . 134 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C (), h Therefore, the phase space HH of the dynamical system St falls into the ortho- a gonal sum p t ¥ e = Å L r H å k k = 0 2 (), of invariant subspaces. Evidently, the dynamics of the system H St in the sub- L space k is completely determined by the system of three ordinary differential equations (7.11)–(7.13).

Lemma 7.1. Assume that ab, Î H and 2 p 1 C = ax()dx < 0 . (7.14) 0 2p ò 0 (), Then the dynamical system H St is dissipative.

Proof. = (),, Let us fix Ny and then consider the initial conditions 0 u0 v0 w0 from the subspace N N = Å L = Å {},, HN å k å Lin e3 k e3 k +1 e3 k +2 , k = 0 k = 0 {} where en are defined by equations (7.8). It is clear that HN is positively invariant and the trajectory N () ==()(); (); () ()(); (); () yt ut v t w t å uk t vk t wk t coskx k = 0 of the system is a function satisfying (7.1)–(7.4) in the classical sense. Let p be the orthoprojector in Hkx onto the span of elements {cos: n = 0,, 1 ¼ , m m}. We introduce a new variable u ()t = ut()+ am instead of the function () am = ()- ut. Here pm p0 a . Equations (7.1)–(7.3) can be rewritten in the form ì - m = - - + ()- + am - mam (7.15) ï ut uxx v * vw* w u * 1 xx , ï í v - m v = v * u +++v * ()q a w * bv* ()p a , (7.16) ï t xx m 0

ï w - m w = w u - v b ++w ()q a w ()p a , î t xx * * * m * 0 (7.17) = - where qm 1 pm . The properties of the convolution operation (see Exer- cises 7.1 and 7.2) enable us to show that Simplified Model of Appearance of Turbulence in Fluid 135

1 d ()mu 2 ++v 2 w 2 + ()u 2 ++v 2 w 2 = 2 dt x x x = ()ma()- + am , ++()m, ()(), + u * 1 u x ux v * qm a v +++()(), ()(), ()(), w * qm a w v * p0 a v w * p0 a w . (7.18)

It is clear that

2 1 2 ()()-u + am * 1, u = -[]C ()u ,,()v * ()p a , v = C ()a []C ()v 0 0 2p 0 0 () ()Î where C0 f is the zeroth Fourier coefficient of the function fx H . Moreo- ver, the estimate £ ³ qmvvx ,,m 1 ³ ()¤ p £ m¤ holds. We choose m 1 such that 12 qma 2 . Then equation (7.18) implies that æö æö d u 2 ++v 2 w 2 ++m u 2 ++v 2 w 2 dt èøèøx x x

++()2 2 ()æö()2 + ()2 £ mam 2 2 C0 u p C0 a èøC0 v C0 w x .

If we use the inequality 2 £ ()2 + 2 f C0 f fx , then we find that

d ()nu 2 ++v 2 w 2 + ()mau 2 ++v 2 w 2 £ m 2 , dt x nm= (),, ¤ p () where min 22C0 a . Consequently, the estimate m ()2 £ ()2 -n t + ()a- -n t m 2 y t H y 0 H e n 1 e x (7.19)

()= ()(),,() () £ mp¤ = + is valid for y t u t vt wt , provided qma 2 and y0 y0 + ()am,, Î 00 where y0 HN . By passing to the limit we can extend inequality Î (), (7.19) over all the elements y0 H . Thus, the system H St possesses an absorbing set B = {}(),, + ()- 2 ++2 2 £ 2 m uvw: upm p0 a v w Rm ,(7.20) - £ mp¤ where map is such that ma 2 and - æöæö1 R2 = m []()p - p a 2 min m,,2 2 C ()a + 1 . m m 0 x èøèøp 0

B Exercise 7.7 Show that the ball m defined by equality (7.20) is positively invariant. 136 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C (), h Exercise 7.8 Consider the restriction of the dynamical system H St to a p the subspace t ìü2 p e ïï r ==()= ()(); (); () Î , () = L H íýhx ux vx wx H ò hxdx 0 å n . ïï³ 2 îþ0 n 1

(), Show that H St is dissipative not depending on the validity of condition (7.14).

Lemma 7.2.

Assume that the hypotheses of Lemma 7.1 hold and let pm be the orthoprojector onto the span of elements {}coskx: k = 01,,¼ ,m in H , = - qm 1 pm . Then the estimate ìüq a ()2 £ 2 - æö()+ 2 m - m ym t y0 expíý2 èøm 1 t (7.21) H H îþ2p < m p ()+ 2 ()= ( (); holds for all mq such that ma 2 m 1 . Here ym t qmut ();)() ()(); (); () qmvt qmwt and ut v t w t is the mild solution to problem = (); ; (7.1)–(7.4) with the initial condition y0 u0 v0 w0 .

Proof. = (); ; Î As in the proof of Lemma 7.1 we assume that y0 u0 v0 w0 HN for some Nq . If we apply the projector m to equalities (7.15)–(7.17), then we get the equations

ì m - m m = - m m - m m ï ut uxx v * v w * w , ï m m í v - m v = vm* um ++wm* bvm* q a , ï t xx m ï m - m m = m m - m + m î wt wxx w * u v * bw* qm a , m = m = m = where u qmu , v qmv and w qmw . Therefore, we obtain æöæö 1 d um 2 ++vm 2 wm 2 + m um 2 ++vm 2 wm 2 £ 2 dt èøèøx x x

£ 1 æöm 2 + m 2 qmavèøw (7.22) 2p as in the proof of the previous lemma. It is easy to check that ()q h 2 ³ m x ³ ()+ 2 2 Î m 1 h for every hpN H . Therefore, inequality (7.22) implies that 1 d ()2 + æöm ()+ 2 - 1 ()2 £ ym t èøm 1 qma ym t 0 . 2 dt H 2p H Hence, equation (7.21) is valid. Lemma 7.2 is proved. Simplified Model of Appearance of Turbulence in Fluid 137

Lemmata 7.1 and 7.2 enable us to prove the following assertion on the existence of the global attractor.

Theorem 7.1. Let ab, Î H and let condition (7.14) hold. Then the dynamical system (), H St generated by the mild solutions to problem (7.1)–(7.4) possesses a global attractor Am . This attractor is a compact connected set. It lies in the finite-dimensional subspace N = Å {},, HN å Lin e3 k e3 k + 1 e3 k + 1 , k = 0 {} where the vectors en are defined by equalities (7.8) and the parameter N < is defined as the smallest number possessing the property qN a < m p ()+ 2 2 N 1 . Here qN is the orthoprojector onto the subspace generated by the elements {}cosnx: n ³ N +1 in H .

To prove the theorem it is sufficient to note that the dynamical system is compact (see Lemma 4.1). Therefore, we can use Theorem 1.5.1. In particular, it should be A H noted that belonging of the attractor m to the subspace N means that dimf A m £ 3 ()N +1 , where N is an arbitrary number possessing the property < m p ()+ 2 qN a 2 N 1 . Below we describe the structure of the attractor and evaluate its dimension exactly.

According to Lemma 7.2 the subspace HN is a uniformly exponentially attracting and positively invariant set. Therefore, by virtue of Theorem 1.7.4 it is sufficient to study the structure of the global attractor of the finite-dimensional dynamical sys- (), tem HN St . To do that it is sufficient to study the qualitative behaviour of the tra- L ££ jectory in each invariant subspace k , 0 kN (see Exercise 7.6). This behaviour is completely described by equations (7.11)–(7.13) which get trans- mm= 2 + d formed into system (1.6.4)–(1.6.6) studied before if we take k k 0 , nm=b2 - = k ak , and bk . Therefore, the results contained in Section 6 of Chapter 1 lead us to the following conclusion.

Theorem 7.2. Let the hypotheses of Theorem 7.1 hold. Then the global minimal attrac- A ()m (), tor min of the dynamical system H St generated by mild solutions to A()m = {}È S problem (7.1)–(7.4) has the form min 0 m , where ìücoskx S = íý()u ; r cosj ; r sinj . m È È k k k p £ j < pîþ kJÎ m ()a 0 2 = m 2 - = ()mm- 2 2 12¤ = () Here uk k ak , rk kak k , the values ak Ck a are the 138 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C () h Fourier coefficients of the function ax, and the number k ranges over 2 a the set of indices Jm()a such that 0

It should be noted that appearance of a limit invariant torus of high dimension pos- sesssing the structure described in Theorem 7.2 is usually assosiated with the Land- au-Hopf picture of turbulence appearance in fluids. Assume that the parameter m gradually decreases. Then for some fixed choice of the function ax() the following picture is sequentially observed. If m is large enough, then there exists only one attracting fixed point in the system. While m decreases and passes some critical value m 1 , this fixed point loses its stability and an attracting limit cycle arises in the system. A subsequent decrease of m leads to the appearance of a two-dimensional mm<

§8§ 8 On Retarded Semilinear Parabolic Equations

In this section we show how the above-mentioned ideas can be used in the study of the asymptotic properties of dynamical systems generated by the retarded perturbations of problem (2.1). It should be noted that systems corresponding to ordinary retarded differential equations are quite well-studied (see, e. g., the book by J. Hale [4]). However, there are only occasional journal publications on the retarded partial differential equations. The exposition in this section is quite brief. We give the reader an opportunity to restore the missing details independently. As before, let A be a positive operator with discrete spectrum in a separable Hilbert space HCab and let (),,q be the space of strongly continuous functions q on the segment []ab, with the values in q =qDA() , ³ 0 . Further we also use the notation Cq = Cr()- , 0 ; q , where r > 0 is a fixed number (with the meaning of the delay time). It is clear that Cq is a Banach space with the norm

ìüq v º max íýA v()s : s Î []-r ; 0 . Cq îþ On Retarded Semilinear Parabolic Equations 139

Let BC be a (nonlinear) mapping of the space q into H possessing the property Bv()- Bv()£ MR()v - v ,0 £ q < 1 , (8.1) 1 2 1 2 Cq for any v , v ÎC such that v £ R , where R > 0 is an arbitrary number and MR() 1 2 q j Cq is a nondecreasing function. In the space H we consider a differential equation u d + Au = Bu(),tt³ , (8.2) dt t 0 where ut denotes the element from Cq determined with the help of the function ut() by the equality ()s =s()+ s Î []- , ut ut ,r 0 . We equip equation (8.2) with the initial condition u ()s ==ut()+ s v ()s , sÎ []-r, 0 , (8.3) t0 0 where vC is an element from q . The simplest example of problem (8.2) and (8.3) is the Cauchy problem for the nonlinear retarded diffusion equation: ¶ ì u - D = ()(), - ()()- , Î W > ï ¶ u f1 ut x f2 ut r x , x , tt0 , í t (8.4) ï ==()s ()s u ¶W 0, u s Î []- , v . î t0 rt0 () () Here rf is a positive parameter, 1 u and f2 u are the given scalar functions. As in the non-retarded case (see Section 2), we give the following definition. ()Î ()- , + ; A function ut Ct0 r t0 T q is called a mild (in q ) solution [ ,)+ () to problem (8.2) and (8.3) on the half-interval t0 t0 T if (8.3) holds and ut satisfies the integral equation t -()tt- A -()tt- A ()= 0 ()+ 0 ()t ut e v 0 òe But d .(8.5)

t0 The following analogue of Theorem 2.1 on the local solvability of problem (8.2) and (8.3) holds.

Theorem 8.1.

Assume that (8.1) holds. Then for any initial condition vCÎ q there existsT > 0 such that problem (8.2) and (8.3) has a unique mild solution [ ,)+ on the half-interval t0 t0 T .

Proof. As in Section 2, we use the fixed point method. For the sake of simplicity we = Î consider the case t0 0 (for arbitrary t0 R the reasoning is similar). In the space Cq ()0, T º C ()0, T ; q we consider a ball 140 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h B = {}wCÎr()0, T : wv- £ , r q Cq ()0, T a p ()= {}- () ()s Î t where v t exp At v 0 and v Cq is the initial condition for problem e (8.2) and (8.3). We also use the notation r w º max {}Aqwt() : t Î []0, T . 2 Cq()0, T

We consider the mapping KC from q()0, T into itself defined by the formula t []()= ()+ -()t - t A ()t Kw t v t òe Bwt d . 0 Here we assume that w()s =sv ()s for Î [-r,)0 . Using the estimate (see Exer- cise 1.23) q q -sA £ æöq > A e èøes ,,s 0 (8.6) (for q =q0 we suppose q = 1 ) we have that t æöq q Aq()Kw ()t - Kw ()t £ Bw()-tBw()d .(8.7) 1 2 òèøet()- t 1, t 2, t 0 If w Î B r , then max Aqwt() £ r + Aq v()0 . []0, T This easily implies that

wt £ r + v ,,t Î []0, T Cq Cq

where, as above, wt Î Cq is defined by the formula

wt ()s =sw ()ts+ ,.Î []-r, 0 Î B Hence, estimate (8.1) for wj r gives us that

Bw(), t - Bw(), t £ M ()r + v w , t - w , t . 1 2 Cq 1 2 Cq ()s =s()s Î [- ,) Since w1 w2 for r 0 , the last estimate can be rewritten in the form

Bw(), t - Bw(), t £ M ()r + v w - w 1 2 Cq 1 2 Cq ()0, T for t Î []0, T . Therefore, (8.7) implies that æöq q Kw - Kw £ 1 T 1- q M ()r + v w - w , 1 2 Cq ()0, T 1 - q èøe Cq 1 2 Cq ()0, T Î B = , if wj r , j 12 . Similarly, we have that q ìü - £ 1 æöq 1- q ()+ ()rr + ()+ Kw v C ()0, T èøT íýB 0 M v C v C q 1 - q e îþq q On Retarded Semilinear Parabolic Equations 141 for w ÎB . These two inequalities enable us to choose TT= ()qr,,v > 0 r Cq such that K is a contractive mapping of Br into itself. Consequently, there exists a unique fixed point wt()Î Cq ()0, T of the mapping K . The structure of the operator Kw implies that ()+ 0 ==[]+Kw ()0 v ()0 . Therefore, the function

ì wt(), t Î []0, T , ut()= í î vt(), trÎ []- , 0 , lies in Cr()- , T ; q and is a mild solution to problem (8.2), (8.3) on the segment []0, T . Thus, Theorem 8.1 is proved.

In many aspects the theory of retarded equations of the type (8.2) is similar to the corresponding reasonings related to the problem without delay (see (2.1)). The exercises below partially confirm that.

Exercise 8.1 Prove the assertions similar to the ones in Exercises 2.1–2.5 and in Theorem 2.2.

Exercise 8.2 Assume that the constant MR() in (8.1) does not depend onR . Prove that problem (8.2) and (8.3) has a unique mild solution [ ,)¥ ()s Î on t0 , provided v Cq . Moreover, for any pair of solu- () () tions u1 t and u2 t the estimate a ()tt- u ()t - u ()t £ a e 2 0 v - v (8.8) 1 2 q 1 1 2 Cq ()s () is valid, where vj is the initial condition for uj t (see (8.3)).

For the sake of simplicity from now on we restrict ourselves to the case when the mapping B has the form ()= ()()+ () = ()s Î Bv B0 v 0 B1 v ,vv C12¤ ,(8.9) (). º ()12 (). where B0 is a continuous mapping from 12¤ DA into HB , 1 continu- ()= ously maps C12¤ into HB and possesses the property 1 0 0 . We also assume (). that B0 is a potential operator, i.e. there exists a continuously Frechét differenti- () ()= - ¢() able function Fu on 12¤ such that B0 u F u . We require that Fu()³ - a,(8.10)()bF ¢()u , u -gFu()³ u 2 - d Îabgd b g for all u 12¤ , where , , , and are real parameters, and are positive () (cf. Section 2). As to the retarded term B1 v , we consider the uniform estimate 0 ()- ()£ 12 ()s()s - ()s B1 v1 B2 v2 MAò v1 v2 d (8.11) -r ()s Î º to be valid. Here Mv is an absolute constant and j CC12¤ . 142 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h Exercise 8.3 Assume that conditions (8.9)–(8.11) hold. Then problem (8.2) a p and (8.3) has a unique mild solution (in 12¤ ) on any segment t []t , t + T for every initial condition vCC from = ¤ . e 0 0 12 r Therefore, we can define an evolutionary operator S acting in the space CCº 2 t 12¤ by the formula ()s()=s()s º ()+ s Î []- , St v ut ut ,,(8.12)r 0 where ut() is a mild solution to problem (8.2) and (8.3).

Exercise 8.4 Prove that the operator St given by formula (8.12) satisfies = = , t ³ the semigroup property: St ° St St + t , S0 I , t 0 and the (), pair C12¤ St is a dynamical system.

Theorem 8.2. Let conditions (8.9)–(8.11) and (8.1) with q = 12¤ hold. Assume that the parameters in (8.10) and (8.11) satisfy the equation

2 ìü r ()2 + g M2 expíýr × min ()2,,gb £ min () 2,,gb. 2g îþ (), Then the dynamical system CSt generated by equality (8.12) is a dissipative compact system.

Proof. We reason in the same way as in the proof of Theorem 4.3. Using the Galerkin approximations it is easy to find that a solution to problem (8.2) and (8.3) satisfies the equalities

1 d ut()2 ++A12 ut()2 ()F ¢()ut(), ut() = ()B ()u , ut() 2 dt 1 t and \ \ 1 d ()A12 ut()2 + 2 Fut()() + u()t 2 = ()B ()u , u()t . 2 dt 1 t If we add these two equations and use (8.10), then we get that \ d Vut()() ++++A12 ut()2 g ut()2 u()t 2 b Fut()() £ dt £ d + ()()() + \ () B1 ut ut u t , (8.13) where

Vu()= 1 u 2 +++1 A12 u 2 Fu() a.(8.14) 2 2 > Using (8.11) it is easy to find that there exists a constant D0 0 such that On Retarded Semilinear Parabolic Equations 143

t d 1 2 Vut()w() + Vut()() £ D + w A12u()t dt , dt 1 0 2 2 ò tr- where æö w =wmin ()2,,gb,.= r 1 + 2 M2 1 2 2 èøg Consequently, the inequality t y\ ()+ wy() £ + w yt()t t 1 t D0 2 ò d tr- is valid for y ()t = Vut()() . Therefore, we have t w t w r j\ () £ 1 + w 1 jt()t t D0 e 2 e ò d tr- for the function w t w t j ()t ==e 1 y ()t e 1 Vut()() . If we integrate this inequality from 0 to t , then we obtain t D w t w r j ()£ j()++0 ()w1 - 1 jt()t t 0 w e 1 2 e r ò d . 1 -r Therefore, Gronwall’s lemma gives us that -w t Vut()() £ ()Vu()()0 + C v 2 e 3 + C , 0 C12¤ 1 provided that w r w 1 < w 2 e r 1 . w r w = w - w 1 > Here C1 and C2 are positive numbers, 3 1 2 e r 0 . This implies (), the dissipativity of the dynamical system CSt . In order to prove its compactness we note that the reasoning similar to the one in the proof of Lemma 4.1 enables us to prove the existence of the absorbing set Bq which is a bounded subset of the q space Cq = Cr()- , 0 ; DA() for 12¤q<<1 . Using the equality (cf. (8.5)) t ()= -()ts- A ()+ -()t - t A ()t ut e us òe But d s for ts³ large enough, we can show that the equation b A12 ()ut()- us() £ Ct- s holds for the solution ut() in the absorbing set Bq . Here the constant C depends on Bq and the parameters of the problem only, b > 0 . This circumstance enables 144 Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations

C h us to prove the existence of a compact absorbing set for the dynamical system a ()CS, . Theorem 8.2 is proved. p t t e Theorem 8.2 and the results of Chapter 1 enable us to prove the following assertion r on the attractor of problem (8.2) and (8.3). 2 Theorem 8.3. Assume that the hypotheses of Theorem 8.2 hold. Then the dynamical (), A system CSt possesses a compact connected global attractor which is a bounded set in the space Cq =qCr()- , 0 ; q for each < 1 .

It should be noted that the finite dimensionality of this attractor can be proved in some cases. References

1. BABIN A. V., VISHIK M. I. Attractors of evolution equations.–Amsterdam: North-Holland, 1992. 2. TEMAM R. Infinite-dimensional dynamical systems in Mechanics and Physics. –New York: Springer, 1988. 3. HENRY D. Geometric theory of semilinear parabolic equations. Lect. Notes in Math. 840. –Berlin: Springer, 1981. 4. HALE J. K. Theory of functional differential equations. –Berlin; Heidelberg; New York: Springer, 1977.

Chapter 3

Inertial Manifolds

Contents

. . . . References ...... 214

If an infinite-dimensional dynamical system possesses a global attractor of finite dimension (see the definitions in Chapter 1), then there is, at least theoretically, a possibility to reduce the study of its asymptotic regimes to the investigation of properties of a finite-dimensional system. However, as the structure of attractor cannot be described in details for the most interesting cases, the constructive investigation of this finite-dimensional system cannot be carried out. In this respect some ideas related to the method of integral manifolds and to the reduction principle are very useful. They have led to appearance and intensive use of the concept of inertial manifold of an infinite-dimensional dynamical system (see [1]–[8] and the references therein). This manifold is a finite-dimensional invariant surface, it contains a global attractor and attracts trajectories exponentially fast. Moreover, there is a possibility to reduce the study of limit regimes of the original infinite-dimensional system to solving of a similar problem for a class of ordinary differential equations. In this chapter we present one of the approaches to the construction of inertial manifolds (IM) for an evolutionary equation of the type: u d + Au = Bu(), t,.u = u (0.1) dt t = 0 0 Here ut() is a function of the real variable t with the values in a separable Hilbert space H . We pay the main attention to the case when A is a positive linear operator with discrete spectrum and Bu(), t is a nonlinear mapping of H subordinated to A in some sense. The approach used here for the construction of inertial manifolds is based on a variant of the Lyapunov-Perron method presented in the paper [2]. Other approaches can be found in [1], [3]–[7], [9], and [10]. However, it should be noted that all the methods for the construction of IM known at present time require a quite l - l strong condition on the spectrum of the operator A : the difference N + 1 N of two neighbouring eigenvalues of the operator A should grow sufficiently fast asN ® ¥ .

§ 1 Basic Equation and Concept of Inertial Manifold

In a separable Hilbert space H we consider a Cauchy problem of the type u d + Au = Bu(), t,,ts> u = u ,,(1.1)s Î R dt ts= 0 where A is a positive operator with discrete spectrum (for the definition see Section 1 of Chapter 2) and B ().., is a nonlinear continuous mapping from DA()q ´ R 150 Inertial Manifolds

C £ q < h intoH , 0 1, possessing the properties a q p Bu(), t £ M ()1 + A u (1.2) t e and r (), - (), £ q ()- Bu1 t Bu2 t MA u1 u2 (1.3) 3 = ()q q for all uu , 1 , and u2 from the domain q DA of the operator A . Here M is a positive constant independent of t and . is the norm in the space H . Further it {} is assumed that ek is the orthonormal basis in H consisting of the eigenfunctions of the operator A : = l () () for 0 1 and ts . Moreover, for any pair of mild solutions u1 t and u2 t to problem (1.1) the following inequalities hold (see (2.2.15)): a ()ts- q () £ 2 q () ³ A ut a1 e A us ,(1.5)ts and (cf. (2.2.18))

ìü-l ()- ()- q N + 1 ts -q1 + a2 ts q Q A ut() £ íýe + M ()1 + k a l + e A us(),(1.6) N îþ1 N 1 ()= ()- () ql where ut u1 t u2 t , a1 and a2 are positive numbers depending on , 1 , = - and MQ only. Hereinafter N IPN , where PN is the orthoprojector onto the first NA eigenvectors of the operator . Moreover, we use the notation ¥ -q -x k =qqq ò x e dx for> 0 andk =q0 for= 0 . (1.7) 0 Further we will also use the following so-called dichotomy estimates proved in Lemma 1.1 of Chapter 2: l t q -tA £ lq N Î A e PN N e ,;t R

- -l t tA £ N + 1 ³ e QN e ,;t 0 (1.8) Basic Equation and Concept of Inertial Manifold 151

-l t q -tA £ []()q¤ q + lq N +1 >q> A e QN t N + 1 e ,,.t 0 0 The inertial manifold (IM) of problem (1.1) is a collection of surfaces {}, Î Mt t R in H of the form = {}+ # (), Î , # (), Î ()- Mt p pt: pPN H pt 1 PN q , #(), ´ ()- where pt is a mapping from PN H R into 1 PN q satisfying the Lipschitz condition q()#()#, - (), £ q()- A p1 t p2 t CA p1 p2 (1.9) with the constant Cp independent of j and t . We also require the fulfillment of the Î () invariance condition (if u0 Ms , then the solution ut to problem (1.1) posses- ()Î ³ ses the property ut Mt , ts ) and the condition of the uniform exponential attraction of bounded sets: there exists g > 0 such that for any bounded set BHÌ > there exist numbers CB and tB s such that ìü-g ()tt- ()(), , Î £ B sup íýdist ut u Mt : u B C e îþq 0 0 B ³ (), for all ttB . Here ut u0 is a mild solution to problem (1.1). From the point of view of applications the existence of an inertial manifold (IM) ()- means that a regular separation of fast (in the subspace IPN H ) and slow (in the subspace PN H ) motions is possible. Moreover, the subspace of slow motions turns out to be finite-dimensional. It should be noted in advance that such separation is not unique. However, if the global attractor exists, then every IM contains it. When constructing IM we usually use the methods developed in the theory of integral manifolds for central and central-unstable cases (see [11], [12]). If the inertial manifold exists, then it continuously depends on t , i.e. limAq()#()#ps, - ()pt, = 0 ts® Î Î () for any pPN H and s R . Indeed, let ut be the solution to problem (1.1) with = + #(), Î ()Î ³ u0 p ps, pPN H . Then ut Mt for ts and hence ()= ()+ #()(), ut PN ut PN ut t . Therefore, #()#, -#(), = []()#, - ()(), + pt ps pt PN ut t ++[]()- []- () ut u0 pPN ut . Consequently, Lipschitz condition (1.9) leads to the estimate q()#()#, - (), £ q()()- A ps pt CA ut u0 . Since ut()Î Cs(),,+¥ DA()q , this estimate gives us the required continuity property of #()pt, . 152 Inertial Manifolds

C h Exercise 1.1 Prove that the estimate a b qb- p A ()#()#pt, +s - ()pt, £ Cb()spN, t e #(), ££s ££bq Î r holds for pt when 0 1 , 0 , t R .

3 The notion of the inertial manifold is closely related to the notion of the inertial form. If we rewrite the solution ut() in the form ut()= pt()+ qt() , where ()= () ()= () = - pt PN ut, qt QN ut , and QN IPN , then equation (1.1) can be rewritten as a system of two equations ì ï d pt()+ Ap() t = P Bpt()()+ qt() , ï dt N ï í d ()+ ()= ()()+ () ï qt Aq t QN Bpt qt , ï dt ï p = p º P u , q = q º Q u . î ts= 0 N 0 ts= 0 N 0 (), Î By virtue of the invariance property of IM the condition p0 q0 Ms implies that ()(), () Î = # (), ()= #()(), pt qt Mt , i.e. the equality q0 p0 s implies that qt pt t . Therefore, if we know the function # ()pt, that gives IM, then the solution ut() lying in Mt can be found in two stages: at first we solve the problem

d pt()+ Ap() t = P Bpt()()+ #()pt(), t ,,(1.10)p = p dt N ts= 0 and then we take ut()= pt()+ #()pt(), t . Thus, the qualitative behaviour of solutions ut() lying in IM is completely determined by the properties of differential

equation (1.10) in the finite-dimensional space PN H . Equation (1.10) is said to be the inertial form (IF) of problem (1.1). In the autonomous case (Bu(), t º Bu() ) one can use the attraction property for IM and the reduction principle (see Theorem 7.4 of Chapter 1) in order to state that the finite-dimensional IF completely determines the asymptotic behaviour of the dynamical system generated by problem (1.1).

Exercise 1.2 Let #()pt, give the inertial manifold for problem (1.1). Show that IF (1.10) is uniquely solvable on the whole real axis, i.e. ()Î ()-¥¥, ; there exists a unique function pt C PN H such that equation (1.10) holds.

Exercise 1.3 Let pt() be a solution to IF (1.10) defined for all t Î R . Prove that ut()= pt()+ #()pt(), t is a mild solution to problem (1.1) defined on the whole time axis and such that u = p + #()p , t . ts= 0 0 Exercise 1.4 Use the results of Exercises 1.2 and 1.3 to show that if IM {} Î Mt exists, then it is strictly invariant, i.e. for any u Mt and < Î = () st there exists u0 Mt such that uut is a solution to problem (1.1). Basic Equation and Concept of Inertial Manifold 153

In the sections to follow the construction of IM is based on a version of the Lyapunov- Perron method presented in the paper by Chow-Lu [2]. This method is based on the following simple fact.

Lemma 1.1. Let f ()t be a continuous function on R with the values in H such that () £ Î QN ft C ,.t R Then for the mild solution ut() (on the whole axis) to equation

d uAu+ = ft() (1.11) dt to be bounded in the subspace QN q it is necessary and sufficient that t t () = -()ts- A ++-()t - t A ()tt -()t - t A ()tt ut e peò PN f d ò e QN f d (1.12) s -¥ Î for t R , where pP is an element from N Hs and is an arbitrary real number.

We note that the solution to problem (1.11) on the whole axis is a function ut()Î Î C ()R, H satisfying the equation t -()- -()- t ut()= e tsA us+ òe t A f ()tt d s for any s Î R .

Proof. It is easy to prove (do it yourself) that equation (1.12) gives a mild solution to (1.11) with the required property of boundedness. Vice versa, let ut() be () a solution to equation (1.11) such that QN ut q is bounded. Then the func- ()= () tion qt QN ut is a bounded solution to equation d qt()+ Aq() t = Q ft(). dt N Consequently, Lemma 2.1.2 implies that t ()= -()t - t A ()tt qt ò e QN f d . -¥ Therefore, in order to prove (1.12) it is sufficient to use the constant variation formula for a solution to the finite-dimensional equation dp + Ap = P ft(),.pt()= P ut() dt N N Thus, Lemma 1.1 is proved. 154 Inertial Manifolds

C #(), h Lemma 1.1 enables us to obtain an equation to determine the function pt . a (), Indeed, let us assume that Bu t is bounded and there exists Mt with the func- p #(), q#(), £ Î Î t tion pt possessing the property A pt C for all pPN H and t R . e Then the solution to problem (1.1) lying in has the form r Mt 3 ut()= pt()+ #()pt(), t .

It is bounded in the subspace QNH and therefore it satisfies the equation of the form ut()= e-()ts- A p + t t ++-()t - t A ()t()tt , -()t - t A ()t()tt , ()Î òe PN Bu d ò e QN Bu d , t R . (1.13) s -¥ Moreover, s #(), ==() -()s - t A ()t()tt , ps QN us ò e QN Bu d . (1.14) -¥ Actually it is this fact that forms the core of the Lyapunov-Perron method. It is proved below that under some conditions (i) integral equation (1.13) is uniquely Î#(), solvable for any pPNH and (ii) the function ps defined by equality (1.14) gives IM. In the construction of IM with the help of the Lyapunov-Perron method an important role is also played by the results given in the following exercises.

-g ()- Exercise 1.5 Assume that sup {}e st ft() : ts< < ¥ , where g is any ()l , l Î () number from the interval N N +1 and s R . Let ut be a mild solution (on the whole axis) to equation (1.11). Show that ut() possesses the property sup {}e-g ()st- Aqut() < ¥ ts< if and only if equation (1.12) holds for ts< . Hint: consider the new unknown function g ()- wt()= e tsut() instead of ut() .

Exercise 1.6 Assume that ft() is a continuous function on the semiaxis [s,)+¥ with the values in H such that for some g from the interval ()l , l N N +1 the equation -g ()- sup {}e stft(): tsÎ [ ,)+¥ < ¥ holds. Prove that for a mild solution ut() to equation (1.11) on the semiaxis [s,)+¥ to possess the property Integral Equation for Determination of Inertial Manifold 155

-g ()- sup {}e stAqut(): tsÎ [ ,)+¥ < ¥ it is necessary and sufficient that t ()= -()ts- A + -()t - t A ()tt - ut e qeò QN f d s +¥ - -()t - t A ()tt ò e PN f d , (1.15) t ³ ()q where ts and qQ is an element of N DA . Hint: see the hint to Exercise 1.5.

§ 2 Integral Equation for Determination of Inertial Manifold

In this section we study the solvability and the properties of solutions to a class of integral equations which contains equation (1.13) as a limit case. Broader treatment of the equation of the type (1.13) is useful in connection with some problems of the approximation theory for IM. Î < £ ¥ º ()- , For s R and 0 L we define the space Cs Cgq, sLs as the set of continuous functions vt() on the segment []sL- , s with the values in DA()q and such that º {}-g ()st- q () < ¥ v s sup e A ut . tsLÎ[]- , s Here g is a positive number. In this space we consider the integral equation ()= sL, []() - ££ vt Bp v t ,,sL ts (2.1) where s sL, []()= -()ts- A - -()t - t A ()t()tt , + Bp v t e peò PB v d t t + ò e-()t - t A QB()t v()tt , d . sL-

Hereinafter the index NP of the projectors N and QN is omitted, i.e. P is the ortho- {},,¼ = - projector onto Lin e1 eN and Q 1 P . It should be noted that the most significant case for the construction of IM is when L = ¥ . 156 Inertial Manifolds

C h Lemma 2.1. a p Let at least one of two conditions be fulfilled: t qq q e æö1 - q 0 <

Proof. Let us apply the fixed point method to equation (2.1). Using (1.8) it is easy to check (similar estimates are given in Chapter 2) that , , Aq()BsL()v ()t - BsL()v ()t £ p1 1 p2 2

s l ()st- q l ()t - t £ N q()l- ++N ()t - ()t t e A p1 p2 ò N e Mv1 v2 q d t t q æöq q -l ()t - t + + l e N +1 Mv()t - v ()t t £ ò èøt - t N + 1 1 2 q d sL-

l ()st- g ()- £ e N Aq()p - p + ()q ()st, + q ()st, e stv - v , 1 2 1 2 1 2 s

where t q æöq q -()l - g ()t - t q ()st, = M + l e N +1 t (2.7) 1 ò èøt - t N +1 d sL- Integral Equation for Determination of Inertial Manifold 157 and s ()tl - g ()- t (), = lq N t q2 st M ò N e d .(2.8) t Therefore, if the estimate (), + (), £ - ££ q1 st q2 st q ,(2.9)sL ts holds, then , , BsL[]v - BsL[]v £ Aq()p - p + qv - v .(2.10) p1 1 p2 2 s 1 2 1 2 s (), (), Let us estimate the values q1 st and q2 st . Assume that (2.2) is fulfilled. Then it is evident that t (), £ qq ()- t -q t + lq ()- + = q1 st M ò t d M N +1 tsL sL-

qq q = M ()ts- + L 1- q + M l ()ts- + L 1- q N +1 and (), ££lq ()- lq ()- q2 st M N st M N +1 st l ££gl for N N +1 . Therefore, æöqq q q ()st, + q ()st, £ M ()ts- + L 1- q + l L . 1 2 èø1- q N +1 Consequently, equation (2.2) implies (2.9). Now let the spectral condition (2.3) be fulfilled. Then t q q l M q -()l - g ()t - t M + q ()st, £ e N +1 t + N 1 1 ò q d l - g ()t - t N +1 -¥ glgl= + ()l¤ q provided that N . Equation (2.3) implies that N 2 Mq N lies in ()l , l g the interval N N +1 . If we choose the parameter in such way, then we get 158 Inertial Manifolds

C q h M ()l1 + k + q a (), + (), £ N 1 + q1 st q2 st . p l - l - 2M lq 2 t N +1 N q N e r Hence, equation (2.3) implies (2.9). Therefore, estimate (2.10) is valid, provi- 3 ded that the hypotheses of the lemma hold. Moreover, similar reasoning enables us to show that sL, [] £ ++q Bp v s D1 A pqvs , (2.11)

where D1 is defined by formula (2.6). In particular, estimates (2.10) and (2.11) sL, mean that when sL , , and p are fixed, the operator Bp maps Cs into itself (), and is contractive. Therefore, there exists a unique fixed point vs tp . Evi- dently it possesses properties (2.4) and (2.5). Lemma 2.1 is proved.

{}L Lemma 2.1 enables us to define a collection of manifolds Ms by the formula L = {}+ #L(), Î Ms p ps: pPH, where s -()- t #L()ps, = ò e s A QB()t v()t , td º Qv() s; p .(2.12) sL- Here vt()= vt(); p is the solution to integral equation (2.1). Some properties of {}L #L, the manifolds Ms and the function ps are given in the following assertion.

Theorem 2.1. Assume that at lealeassstt one of two conditions (2.2) and (2.3) is satisfied. Then the mapping #L(). , s from PH into QH possesses the properties q#L(), £ + ()- -1 {}+ q a) A ps D2 q 1 q D1 A p (2.13) Î for any pPH , hereinafter D1 is defined by formula (2.6) and = ()l+ -q1 + D2 M 1 k N +1 ; (2.14) L b) the manifold Ms is a Lipschitzian surface and q Aq#L()#p , s - L()p , s £ Aq()p - p (2.15) 1 2 1 - q 1 2 , Î Î for all p1 p2 PH and s R ; ()º (), ; + #L() c) if ut ut s p s p is the solution to problem (1.1) with the = + #L(), Î ()= #L()(), initial data u0 p ps, pPH, then Qu t Pu t t for L = ¥ . In case of L < ¥ the inequality Aq()Qu() t - #L()Pu() t , t £ £ ()- -1 -g L + ()- -2 -g ()ts- {}+ q D2 1 q e q 1 q e D1 A p (2.16) Integral Equation for Determination of Inertial Manifold 159

holds for all stsL££+ , where g is an arbitrary number from the []l , l gl= + ()l¤ q segment N N +1 if (2.2) is fulfilled and N 2 Mq N when (2.3) is fulfilled; d) if Bu(), t º Bu() does not depend on t , then #L()#ps, º L()p , i.e. #L()pt, is independent of t .

Proof. Equations (2.12) and (1.8) imply that s q æöq q -l ()s - t Aq#L()ps, ££M + l e N +1 ()t1 + Aqv()t ò èøs - t N +1 d sL- s q æöq q -l ()s - t £ M + l e N +1 t + q ()ss, v . ò èøs - t N +1 d 1 s sL- (), < By virtue of (2.9) we have that q1 ss q . Therefore, when we change the vari- xl= ()-t able in integration N +1 s with the help of equation (2.5) we obtain (2.13). Similarly, using (2.4) and (1.8) one can prove property (2.15). Î [], + () Let us prove assertion (c). We fix t0 ss L and assume that wt is a [], + ()= () Î [], ()= function on the segment ss L such that wt ut for tst0 and wt = () Î[]- , () vs t for tsLs . Here vs t is the solution to integral equation (2.1). Using equations (1.4) and (2.1) we obtain that t -()- t wt()==e-()ts- A ()p + #L()ps, + òe t A Bw()t()tt , d s t t -()- -()- t -()- t = e tsA pe++ò t A PB()t w()t , td ò e t A QB()t w()tt , d (2.17) s sL- ££ Î []- , for stt0 . Evidently, equation (2.17) also remains true for tsLs . Equa- tion (1.4) gives us that s -()st- A = 0 ()+ -()s - t A ()t()t , t pe pt0 òe PB w d .

t0 Therefore, the substitution in (2.17) gives us that , t0 L wt()= B ()[]w ()t + b ()t , st; (2.18) pt0 L 0 Î []- , ()= () for all tt0 L t0 , where pt Pu t and - t0 L (), ; = -()t - t A ()t()t , t bL t0 st ò e QB vs d .(2.19) sL- 160 Inertial Manifolds

C = ¥ = h In particular, if L equation (2.18) turns into equation (2.1) with st0 and a = () ()= ppt0 . Therefore, equation (2.12) implies the invariance property Qu t0 p = #¥()(), t Pu t0 t0 . Let us estimate the value (2.19). If we reason in the same way as e in the proof of Lemma 2.1, then we obtain that r -()ltt- + L ìüg ()st- + L 3 Aq b ()t , st; £ e 0 N +1 íýq* ()st, -L + q ()st, -L e 0 v , L 0 îþ1 0 1 0 s s (), where q1 st is defined by formula (2.7) and t q æöæöq q -l ()t - t q* ()st, = M + l e N +1 t .(2.20) 1 ò èøèøt - t N +1 d sL- Therefore, simple calculations give us that

-()ltt- + L ìüg ()st- + L Aqb ()t , st; £ e 0 N +1 íýD + e 0 qv ,(2.21) L 0 îþ2 s s where D is defined by formula (2.14). Let v ()t be the solution to integral equa- 2 t0 = = () tion (2.1) for st0 and pPut0 . Then using (2.12), (2.18), and (2.1) we find that Qu() t - #L()Pu() t , t = Qwt()()- v ()t .(2.22) 0 0 0 0 t0 0 Î []- , However, for all tt0 L t0 we have that t , L t , L ()- ()= 0 []()- 0 []()+ (), ; wt v t Bpt()w t Bpt()v t b t st. t0 0 0 t0 L 0 , t0 L Therefore, the contractibility property of the operator Bp gives us that ìü-g ()t - t ()1 - q wv- £ sup íýe 0 Aqb ()t , st; . t0 t Î []- , L 0 0 tt0 Lt0 îþ Hence, it follows from (2.21) and (2.22) that

Aq()#Qu() t - L()Pu() t , t ££Aq()wt()- v ()t 0 0 0 0 t0 0

- ìü-g ()t - s ££wv- ()1 - q 1 íýe-g L D + qe 0 v . t 2 s s 0 t0 îþ This and equation (2.5) imply (2.16). Therefore, assertion (c) is proved. In order to prove assertion (d) it should be kept in mind that if * ()ºut, º * () sL, u , then the structure of the operator Bp enables us to state that sL, []()- = shL+ , []() Bp v th Bp vh t + - ££+ ()= ()- ()Î ()- , for shL t sh , where vh t vt h . Therefore, if vt Cgq, sLs is a solution to integral equation (2.1), then the function () º ()- Î ()+ - , + vh t vt h Cgq, shLsh Existence and Properties of Inertial Manifolds 161 is its solution when sh+ is written instead of s . Consequently, equation (2.12) gives us that #L(), + ===()+ () #L(), ps h Qvh sh Qv s ps . Thus, Theorem 2.1 is proved.

Exercise 2.1 Show that if Bu(), t £ M , then inequalities (2.13) and (2.16) can be replaced by the relations q#L(), £ A ps D2 , (2.23) q()()- #L()()(), £ ()- -1 -g L A Qu t Pu t t D2 1 q e , (2.24)

where D2 is defined by formula (2.14).

§ 3 Existence and Properties of Inertial Manifolds

In particular, assertion (c) of Theorem 2.1 shows that if the spectral gap condition l - l ³ 2M ()()l+ q + lq << N +1 N q 1 k N +1 N ,,(3.1)0 q 1 is fulfilled, then the collection of surfaces = {}+ #(), Î Î Ms p ps: pPH,,s R (3.2) is invariant, i.e. (), Ì¥- < £ < ¥ Ut sMs Mt ,.st (3.3) Here #()#ps, = ¥()ps, is defined by formula (2.12) for L = ¥ and Ut(), s is the evolutionary operator corresponding to problem (1.1). It is defined by the for- (), = () () mula Ut su0 ut , where ut is a mild solution to problem (1.1). In this section we show that collection (3.2) possesses the property of exponen- {} tial uniform attraction. Hence, Mt is an inertial manifold for problem (1.1). More- over, Theorem 3.1 below states that {}M is an exponentially asymptotically t ()= (), ()= complete IM, i.e. for any solution ut Ut su0 there exists a solution u t = (), () ()Î , ³ Ut su0 lying in the manifold u t Mt ts such that -h ()- Aq()ut()- u ()t £ Ce ts,,.h > 0 ts>

In this case the solution u ()t is said to be an induced trajectory for ut() on the manifold Mt . In particular, the existence of induced trajectories means that the solution to original infinite-dimensional problem (1.1) can be naturally associated with the solution to the system of ordinary differential equations (1.10). 162 Inertial Manifolds

C h Theorem 3.1. a < - p Assume that spectral gap condition (3.1) is valid for some q 22. t {}, Î Then the manifold Ms s R given by formula (3.2) is inertial for prob- e ()= (), r lem (1.1). Moreover, for any solution ut Ut su0 there exists an induced trajectory u*()t = Ut(), su* such that u*()t Î for ts³ and 3 duced trajectory 0 such that Mt for and 21()- q -g ()- Aq()ut()- u*()t £ e tsAq()Qu - #()P us, , (3.4) ()2 - q 2 - 2 0 0 gl= + 2 M lq ³ where N q N and ts. Proof. Obviously it is sufficient just to prove the existence of an induced trajectory * ()Î () u t Mt possessing property (3.4). Let ut be a mild solution to problem (1.1), ()= (), *()= (), * () ut Ut su0 . We construct the induced trajectory u t Ut su0 for ut *()= ()+ () () + º ( ,,+¥ in the form u t ut wt , where wt lies in the space Cs Cs, g s DA())q of continuous functions on the semiaxis [s,)+¥ such that º {}g ()ts- q () < ¥ w s, + sup e A wt ,(3.5) ts³ gl= + ()l¤ q where N 2 Mq N . We introduce the notation Fw(), t = But()()+ w, t - But()() (3.6) and consider the integral equation (cf. (1.15)) t ()= + []()º -()ts- A ()+ -()t - t A ()t()tt , - wt Bs w t e qw òe QF w d s +¥ -()t - t A - ò e PF()t w()tt , d , tsÎ [ ,)+¥ , (3.7) t + ()Î ()q in the space Cs . Here the value qw QD A is chosen from the condition *()= ()+ ()Î u s us ws Ms , i.e. such that +#()= ()+ (), Qu0 Qw s Pu0 Pw s s . Therefore, by virtue of (3.7) we have æö+¥ ()= -#+ ç÷- -()s - t A ()t()tt , , qw Qu0 ç÷Pu0 ò e PF w d s .(3.8) èøs

Thus, in order to prove inequality (3.4) it is sufficient to prove the solvability of inte- + gral equation (3.7) in the space Cs and to obtain the estimate of the solution. The preparatory steps for doing this are hidden in the following exercises. Existence and Properties of Inertial Manifolds 163

Exercise 3.1 Assume that Fw(), t has the form (3.6). Show that for any (), ()Î + = (), +¥; ()q wt wt Cs Cs, g s DA and for ts³ the following inequalities hold: ()(), £ -g ()ts- Fwt t e Mws, + ,(3.9) ()(), - ()(), £ -g ()ts- - Fwt t Fwt t e Mw ws, + .(3.10)

Exercise 3.2 Using (1.8) prove that the equations +¥ æölq ç÷-()- t -gt()- N -g ()- Aqe t A Pe s t £ × e ts,(3.11) ç÷ò d gl- N èøt æöt ç÷q -()t - t A -gt()- s t £ ç÷ò A e Qe d èøs ()l - g q + lq k + + -g ()- £ N 1 N 1 × e ts (3.12) l - g N +1 l <

Lemma 3.1. Assume that spectral gap condition (3.1) holds with q < 22- . Then + + Bs is a continuous contractive mapping of the space Cs into itself. The unique fixed point w of this mapping satisfies the estimate

21()- q q w , + £ A ()Qu - #()Pu , s . (3.13) s ()2 - q 2 - 2 0 0

Proof. If we use (3.7), then we find that

-()lts- q + []() £ N +1 q ()+ A Bs w t e A qw t +¥ -()- t -()- t + ò Aqe t A QFw()t()tt , d + ò Aqe t A PFw()t()tt , d s t for ts> . Therefore, (3.9), (3.11), and (3.12) give us that

-()lts- q + []() £ N +1 q ()+ A Bs w t e A qw

q ()l+ q ìül 1 k N +1 -g ()- + íýN + Me tsw . gl- l - g s, + îþN N +1 164 Inertial Manifolds

C gl= + ()l¤ q h Since N 2 Mq N , spectral gap condition (3.1) implies that a + -()lts- -g ()- p Aq B []w ()t £ e N +1 Aqqw()+ qe tsw .(3.14) t s s, + e r Similarly with the help of (3.10)–(3.12) we have that q()+ []()- + []() £ 3 A Bs w t Bs w t -()lts- £ N +1 q()()- ()+ -g ()ts- - e A qw qw qe wws, + (3.15) , Î + for any ww Cs . From equations (3.8), (3.9), and (2.15) we obtain that +¥ qM -()- t -gt()- Aq()qw()+ Qu - #()Pu , s £ Aqe s A Pe s dt × w . 0 0 1 - q ò s, + s Therefore, (3.11) implies that q2 Aqqw() £ Aq()Qu - #()Pu , s + w . 0 0 21()- q s, + Similarly we have that q2 Aq()qw()- qw() £ ww- .(3.17) 21()- q s, + It follows from (3.14)–(3.17) that

+ q 2 - q B []w £ Aq()Qu - #()Pu , s + × w ,(3.18) s s, + 0 0 2 1 - q s, +

+ + q 2 - q B []w - B []w £ × ww- . s s s, + 2 1 - q s, + < - + Therefore, if q 22 , then the operator Bs is continuous and contractive in + Cs . Estimate (3.13) of its fixed point follows from (3.18). Lemma 3.1 is proved.

In order to complete the proof of Theorem 3.1 we must prove that the function u*()t = ut()+ wt() {}, ³ () is a mild solution to problem (1.1) lying in Mt t s (here wt is a solution to integral equation (3.7) ). We can do that by using the result of Exercise 1.2, the in- {} variance of the collection Mt , and the fact that equality (3.8) is equivalent to the *()Î equation u s Ms . Theorem 3.1 is completely proved.

Exercise 3.3 Show that if the hypotheses of Theorem 3.1 hold, then the induced trajectory u*()t is uniquely defined in the following sense: if **() **()Î ³ there exists a trajectory u t such that u t Mt for ts and -g ()- Aq()ut()- u**()t £ Ce ts q with gl³ + 2 M l , then u**()t º u*()t for ts³ . N q N Existence and Properties of Inertial Manifolds 165

The construction presented in the proof of Theorem 3.1 shows that in order to build the induced trajectory for a solution ut() with the exponential order of decrease g given, it is necessary to have the information on the behaviour of the solution ut() for all values ts³ . In this connection the following simple fact on the exponential closeness of the solution ut() to its projection Pu() t + #()Pu() t , t onto the manifold appears to be useful sometimes.

Exercise 3.4 Show that if the hypotheses of Theorem 3.1 hold, then the estimate Aq()Qu() t - #()Pu() t , t £

-g ()- £ 2 e ts Aq()Pu - #()Pu , t ()2 - q 2 - 2 0 0 () g= l + is valid for any solution ut to problem (1.1). Here N + ()l¤ q ³#(()-*(), 2 Mq N and ts (Hint: add the value Pu t t -)Qu*()t = 0 to the expression under the norm sign in the left-hand side. Here u*()t is the induced trajectory for ut() ).

{} () It is evident that the inertial manifold Mt consists of the solutions ut to problem (1.1) which are defined for all real t (see Exercises 1.3 and 1.4). These solutions can be characterized as follows.

Theorem 3.2. Assume that spectral gap condition (3.1) holds with q < 22- and {} Mt is the inertial manifold for problem (1.1) constructed in Theorem 3.1. Then for a solution ut() to problem (1.1) defined for all t Î R to lie in the ()()Î inertial manifold ut Mt , it is necessary and sufficient that º {}-g ()st- q () -¥ < £ < ¥ u s sup e A ut : ts (3.19) Îgl= + 2 M lq for each s R , where N q N . Proof. ()Î ()= ()+ #()(), If ut Mt , then ut Pu t Pu t t . Therefore, equation (2.13) implies that qD 1 Aqut() £ D ++1 AqPu() t .(3.20) 2 1 - q 1 - q The function pt()= Pu() t satisfies the equation t -()- t pt()= e-()ts- A ps()+ òe t A PB()t u()t , td s for all real ts and . Therefore, we have that 166 Inertial Manifolds

C h s ()lst- q ()lt - t a Aqpt() £ e N Aqps()+ M l e N ()t1 + Aqu()t d p N ò t t e r for ts£ . With the help of (3.20) we find that 3 q s ()lst- M l ()lt - t Aqpt() £ Cs(),, N qe N + N e N Aqp()t d t 1 - q ò t for ts£ , where æöqD 1 Cs(),, N q = Aqps()+ M 1 ++D 1 . èø2 1 - q l1 - q N Hence, the inequality q s M l jt £ Cs(),, N q + N jtdt 1- q ò t ()lts- holds for the function jt = Aqpt()e N and ts£ . If we introduce the func- s tion yt = ò jtdt , then the last inequlity can be rewritten in the form t q M l yt + N y()t ³ -Cs(),, N q ,,ts£ 1- q or q q ìüM l ìüM l d y ()t exp íýN t ³ -Cs(),, N qexp íýN t ,.ts£ dt îþ1- q îþ1- q After the integration over the segment []ts, and a simple transformation it is easy to obtain the estimate q ìüM l Aqpt() £ Cs(),, N qexp æöl + N ()- .(3.21) íýèøN st îþ1- q Obviously for q < 22- we have that q M l q l +lN < g = + 2M l . N 1- q N q N Therefore, equations (3.21) and (3.20) imply (3.19). Vice versa, we assume that equation (3.19) holds for the solution ut() . Then ()() £ g ()st- ()+ £ But e M 1 u s ,.ts (3.22) -g ()- It is evident that qt()=¥e stQu() t is a bounded (on (- ,]s ) solution to the equation q d + ()A - g q = Ft(), dt Existence and Properties of Inertial Manifolds 167 where Ft()= exp {}- g()st- QBut()() . By virtue of (3.22) the function Ft() is bounded in QH . It is also clear that Ag = A - g is a positive operator with discrete spectrum in QH . Therefore, Lemma 1.1 is applicable. It gives t Qu() t = ò e-()t - t A QB()t u()t d . -¥ Using the equation for Pu() t it is now easy to find that ()= s, ¥ []() £ ut Bp u t ,,ts = () s, ¥ [] where pPus and Bp u is the integral operator similar to the one in (2.1). Hence, we have that Qu() s = #()Pu() s , s accoring to definition (2.12) of the function #()#ps, = ¥()ps, . Thus, Theorem 3.2 is proved.

º ¥ The following assertion shows that IM Ms Ms can be approximated by the mani- {}L < ¥ folds Ms , L , with the exponential accuracy (see (2.12)).

Theorem 3.3. Assume that spectral gap condition (3.1) is fulfilled with q < 1 . We also assume that the function #L()ps, is defined by equality (2.12) for 0 < L £ ¥ . Then the estimate L L Aq()# 1()#ps, - 2()ps, £

-g L 1+ q -d L £ D ()1- q -1 e N min + {}D + Aqp e N min , (3.23) 2 21()- q 2 1 = (), < £ ¥ is valid with Lmin min L1 L2 , 0 L1 , L2 ; the constants D1 and D2 are defined by equations (2.6) and (2.14); q 2 M ()1- q q g =dl + 2 M l ,,.= l . N N q N N q ()1+ q N

Proof. <<<¥ Let 0 L1 L2 . Definition (2.12) implies that L L # 1()#, - 2(), = ()()- () ps ps Qv1 s v2 s ,(3.24) where vj()t is a solution to integral equation (2.1) with LL= , j = 12, . The ope- , j sL2 ()- , rator Bp acting in Cgq, sL2 s (see (2.1)) can be represented in the form sL, sL, 2 []()= 1 []()+ (); , Î []- , Bp v t Bp v t bv t s,,tsL1 s where - sL1 bv(); t, s = ò e-()t - t A QB()t v()tt , d - sL2 168 Inertial Manifolds

C () ()- , () h and vt is an arbitrary element in Cgq, sL2 s . Therefore, if vj t is a solution a to problem (2.1) with LL= , then p j t sL, sL, ()- ()= 1 []- 1 []- (); , e v1 t v2 t Bp v1 Bp v2 bv2 t s (3.25) r for all sL- ££ts . Let us estimate the value bv(); t, s . As before it is easy to 3 1 2 verify that -()lts- + L Aqbv(); t, s £ e 1 N +1 {}r ()sL- + r ()sL- v 2 1 1 2 1 2 s, * Î []- , for all tsL1 s , where t ()= q -()t - t A t r1 t MAò e Q d , -¥ t g ()- t ()= q -()t - t A * s t r2 t MAò e Qe d , -¥ + * =g1q = l + 2M lq and the norm v2 , is defined using the constants q and * s * 2 N q* N by the formula

ìü-g ()st- v = sup íýe * Aqv ()t : tsLÎ []- , s . 2 s, * îþ2 2

Evidently, spectral gap condition (3.1) implies the same equation with the parameter q* instead of q . Therefore, simple calculations based on (1.8) give us that

-g ()st- q* r ()t £ D andr £ e * , 1 2 2 2

where D2 is defined by formula (2.14). Using Lemma 2.1 under condition (2.3) with q* instead of q we obtain that - v £ ()1- q* 1 {}D + Aqp , 2 s, * 1

where D1 is given by formula (2.6). Therefore, finaly we have that -()lts- + L ìü* g q (); , £ 1 N +1 q * L1 q A bv2 t s e íýD + e ()D + A p îþ2 21()- q* 1 Î []- , for all tsL1 s . Consequently,

ìüg ()- supíýe tsAqbv(); t, s : tsLÎ []- , s £ îþ2 1

-g L ìüq* g L £ e 1 íýD + e * 1 ()D + Aqp . îþ2 21()- q* 1 Existence and Properties of Inertial Manifolds 169

sL, 1 ()- , Therefore, since Bp is a contractive operator in Cgq, sL1 s , equation (3.25) gives us that ()- - £ 1 q v1 v2 ()- , Cgq, sL1 s

-g L 1 1+ q -()gg- L æö+ q £ e 1 D + × e * 1 D1 A p . 2 2 1- q èø

Here we also use the equality q* = 1 ()1 + q . Hence, estimate (3.23) follows from 2 (3.24). Theorem 3.3 is proved.

Exercise 3.5 Show that in the case when Bu(), t £ M equation (3.23) can be replaced by the inequality L L -g L q()# 1 ()#, - 2 (), £ ()- -1 N min A ps ps D2 1 q e .

Exercise 3.6 Assume that the hypotheses of Theorem 3.1 hold. Then the estimate Aq()Qu() t - #L()Pu() t , t £ q -g ()ts- -al L £ æö1 + Aqu N + N C èø0 e CR e ³ () holds for tt* and for any solution ut to problem (1.1) possess- q () £ ³³ ing the dissipativity property: A ut R for tt* s and for g =al + 2 M lq > some Rt and * . Here N N q N and the constant 0 does not depend on N .

Therefore, if the hypotheses of Theorem 3.1 hold, then a bounded solution to prob- lq lem (1.1) gets into the exponentially small (with respect to N and L ) vicinity of {}L -¥ <<¥ the manifold Ms : s at an exponential velocity. {}L According to (2.12) in order to build an approximation Ms of the inertial {} manifold Ms we should solve integral equation (2.1) for L large enough. This equation has the same structure both for L < ¥ and for L = ¥ . Therefore, it is im- {}L {} possible to use the surfaces Ms directly for the effective approximation of Ms . s, ¥ – = However, by virtue of contractiveness of the operator Bp in the space Cs = ()-¥, () Cgq, s , its fixed point vs t which determines Ms can be found with the {} help of iterations. This fact enables us to construct the collection Mns, of appro- {} = (); – ximations for Ms as follows. Let v0 v0, s tp be an element of Cs . We take º (), = s, ¥ []() = ,,¼ vn vns, tp Bp vn - 1 t ,,n 12 {} and define the surfaces Mns, by the formula = {}+ # (), Î Mns, p n ps: pPH, # (), = (), = ,,¼ where n ps Qvns, ps , n 12 170 Inertial Manifolds

C º (), º () h Exercise 3.7 Let v0 p and let Bu t Bu . Show that a # (), º#(), = -1 () p 0 ps 0 and1 ps A QB p . t e r Exercise 3.8 Assume that spectral gap condition (3.1) is fulfilled. Show 3 that æö- Aq()# ()#ps, - ()ps, £ qn v + ()1 - q 1 D + Aqp , n èø0 s 1 #(), where D1 is defined by formula (2.6) and ps is the function that determines the inertial manifold. # (), Exercise 3.9 Prove the assertion for n ps similar to the one in Exer- cise 3.5.

Theorems represented above can also be used in the case when the original system is dissipative and estimates (1.2) and (1.3) are not assumed to be uniform with respect to uDAÎ ()q . The dissipativity property enables us to restrict ourselves to the consideration of the trajectories lying in a vicinity of the absorbing set when we study the asymptotic behaviour of solutions to problem (0.1). In this case it is convenient to modify the original problem. Assume that the mapping Bu(), t is continuous with respect to its arguments and possesses the properties (), £ (), - (), £ q()- Bu t Cr ,(3.26)Bu1 t Bu2 t Cr A u1 u2 r > = {}q £ r for any 0 and for all uu , 1 , and u2 lying in the ball Br v : A v . Letc ()s be an infinitely differentiable function on R+ = [0,)¥ such that c ()s = 1 ,;0 ££s 1 c()s = 0 ,;s ³ 2

0 ££c¢c ()s 1 ,,.()s £ 2 sRÎ + (), We define the mapping BR ut by assuming that ()c, = ()-1 q (), Î ()q BR ut R A u Bu t,.(3.27)uDA

(), Exercise 3.10 Show that the mapping BR ut possesses the properties q (), £ A BR ut M , (), - (), £ q()- BR u1 t BR u2 t MA u1 u2 ,(3.28) = ()+ ¤ where MC2 R 12R and Cr is a constant from (3.26).

Let us now assume that Bu(), t satisfies condition (3.26) and the problem u d + Au = Bu(), t,,u = u (3.29) dt t = 0 0 has a unique mild solution on any segment []ss, + T and possesses the following > > dissipativity property: there exists R0 0 such that for any R 0 the relation Continuous Dependence of Inertial Manifold on Problem Parameters 171

q (), ; £ - ³ () A ut s u0 R0 for allts t0 R (3.30) q £ (), ; holds, provided that A u0 R . Here uts u0 is the solution to problem (3.29).

Exercise 3.11 Show that the asymptotic behaviour of solutions to problem (3.29) completely coincides with the asymptotic behaviour of solutions to the problem

du + = (), = Au B2 R ut,,(3.31)u = u0 dt 0 ts where B is defined by formula (3.27) and R is the constant 2 R0 0 from equation (3.30).

Exercise 3.12 Assume that for a solution to problem (3.29) the invariance q £ property of the absorbing ball is fulfilled: if A u0 R0 , then q (), ; £ £ A uts u0 R for all ts . Let Mt be the invariant manifold R0 = Ç {}q £ of problem (3.31). Then the set Mt Mt u : A uR0 is in- Î R0 (), ; Î R0 variant for problem (3.29): if u0 Ms , then uts u0 Ms , ts³ .

Thus, if the appropriate spectral gap condition for problem (3.29) is fulfilled, then there exists a finite-dimensional surface which is a locally invariant exponentially attracting set.

In conclusion of this section we note that the version of the Lyapunov-Perron method represented here can also be used for the construction (see [13]) of inertial manifolds for retarded semilinear parabolic equations similar to the ones considered in Section 8 of Chapter 2. In this case both the smallness of retardation and the fulfilment of the spectral gap condition of the form (3.1) are required.

§ 4 Continuous Dependence of Inertial Manifold on Problem Parameters

Let us consider the Cauchy problem u d + Au = B* ()ut, ,,u = u s Î R (4.1) dt ts= 0 in the space HB together with problem (1.1). Assume that * ()ut, is a nonlinear mapping from DA()q ´ R into H possessing properties (1.2) and (1.3) with the same constant M as in problem (1.1). If spectral gap condition (3.1) is fulfilled, then problem (4.1) (as well as (1.1)) possesses an invariant manifold 172 Inertial Manifolds

C * * h = {}+ # (), Î Î Ms p ps: pPH,.s R (4.2) a p The aim of this section is to obtain an estimate for the distance between the t * e manifolds Ms and Ms . The main result is the following assertion. r 3 Theorem 4.1. Assume that conditions (1.2), (1.3), and (3.1) are fulfilled both for problems (1.1) and (4.1). We also assume that (), -r*(), £ + r q Bv t B vt 1 2 A v (4.3) Î ()q Îrr for all vDA and t R , where 1 and 2 are positive numbers. Then the equation r + r q()#()#, - *(), £ (), q 1 2 + ()r,,q q sup A ps ps C1 q C2 q M 2 A p sRÎ l1 - q N is valid for the functions #()ps, and #*()ps, which give the invariant manifolds for problems (1.1) and (4.1) respectively. Here the numbers (), q (),,q r C1 q and C2 q M do not depend on N and j .

Proof. Equation (2.12) with L = ¥ implies that s -()- t Aq()#()#ps, - *()ps, £ ò Aqe s A QBv()()tt , -tB*()v*()tt , d , -¥ where v()t and v*()t are solutions to the integral equations of the type (2.1) corresponding to problems (1.1) and (4.1) respectively. Equations (1.3) and (4.3) give us that ()()tt , - *()*()tt , ££q()r()t - *()t + æö+ r q ()t Bv B v MA v v èø1 2 A v

£ g ()s - t ()r- * + r + (4.4) e Mv v s 2 v s 1 for t £ s , where ìü = -g ()st- Q () w s ess sup íýe A wt (4.5) ts£ îþ gl= + 2M lq and N q N as before. Hence, after simple calculations as in Section 2 we find that q æör 1 + k Aq()#()#ps, - *()ps, £ vv- * + 2 v + r .(4.6) 2 èøs M s 1 l1 - q N +1 - * * Let us estimate the value vvs . Since vv and are fixed points of the correspon- s, ¥ ding operator Bp , we have that Continuous Dependence of Inertial Manifold on Problem Parameters 173

s -()- t Aq()vt()- v*()t £ ò Aq e t A PBv()()t , t -tB*()v*()tt , d + t s -()- t + ò Aq e t A QBv()()t , t-tB*()v*()tt , d . -¥ Therefore, by using spectral gap condition (3.1) and estimate (4.4) as above it is easy to find that r q 2 æö1 1 + k vv- * £ qv- v* ++ ×rv ç÷ + . s s M s 1 l1- q l1 - q èøN N + 1 Consequently, q r r 2 + k vv- * £ ××2 v + 1 × . s 1- q M s 1- q l1 - q N Therefore, equation (4.6) implies that

q r 2 - q 2 + k Aq()#()#ps, - *()ps, £ ××2 v + r×× . 1- q 2 M s 1 22- q l1- q N Hence, estimate (2.5) gives us the inequality

r + r 2 + k q r Aq()#()#ps, - *()ps, £ 1 2 × + × 2 Aqp . ()1- q 2 l1 - q 21()- q 2 M N This implies the assertion of Theorem 4.1.

() Let us now consider the Galerkin approximations um t of problem (1.1). We remind (see Chapter 2) that the Galerkin approximation of the order m is defined as a () function um t with the values in PmH , this function being a solution to the problem du m + = () = Aum Pm Bum ,.(4.7)um u0 m dt 0 ts= {},,¼ Here Pm is the orthoprojector onto the span of elements e1 em in H .

Exercise 4.1 Assume that spectral gap condition (3.1) holds and mN³ +1 . Show that problem (4.7) possesses an invariant manifold of the form ()m = {}+ #()m (), Î Ms p ps: pPH #()m (), ® ()- in PmH , where the function ps: PH Pm P H is defined by equation similar to (2.12). 174 Inertial Manifolds

C h The following assertion holds. a p Theorem 4.2. t e Assume that spectral gap condition (3.1) holds. Let #()ps, and r #()m ()ps, be the functions defined by the formulae of the type (2.12) and 3 let these functions give invariant manifolds for problems (1.1) and (4.7) for mN³ + 1 respectively. Then the estimate ìü (),,q ïïD + Aqp q()#()#, - ()m (), £ Cq M + 1 A ps ps - q íý1 (4.8) l1 ïïl + m + 1 1 - N 1 îþl m + 1

is valid, where the constant D1 is defined by formula (2.6).

Proof. It is evident that ()#()#ps, - ()m ()ps, = Qvs[](); p - v()m ()sp; ,(4.9) where vt(), p and v()m ()tp, are solutions to the integral equations ()=¥s, ¥ []() - < £ vt Bp v t ,,ts and ()m ()=¥s, ¥ []()m () - < £ v t Pm Bp v t ,.ts s, ¥ Here Bp is defined as in (2.1). Since ()- ()m () = ()- ()+ []s, ¥ []()- s, ¥ []()m () vt v t IPm vt Pm Bp v t Bp v t , we have q()()- ()m () = q()- () + q[]s, ¥ []()- s, ¥ []()m () A vt v t A 1 Pm vt A Bp v t Bp v t . s, ¥ The contractiveness property of the operator Bp leads to the equation q()()- ()m () = q()- () + × - ()m g ()st- A vt v t A 1 Pm vt qv v s e . In particular, this implies that

()m -g ()st- q ()m -1 vv- s º sup e A ()vt()- v ()t £ ()1 - q ()1 - P v . ts< m s Hence, with the help of (4.9) we find that

q ()m q ()m ()m A ()#()#ps, - ()ps, £££A ()vs()- v ()s vv- s (4.10) £ ()1- q -1 ()1 - P v . m s Continuous Dependence of Inertial Manifold on Problem Parameters 175

Let us estimate the value ()1- P v . It is clear that m s t ()- ()= -()t - t A ()- ()t()t 1 Pm vt ò e 1 Pm Bv d . -¥ Therefore, Lemma 2.1.1 (see also (1.8)) gives us that t q æöq q -()lt - t Aq()1 - P vt() £ M + l e m +1 t + m ò èøt - t m +1 d -¥ t q æöq q -l ()t - t ()gt - t g ()- + M + l e m +1 e t v × e st , ò èøt - t m +1 d s -¥ where glº + 2 M lq <

Exercise 4.2 In addition assume that the hypotheses of Theorem 4.2 hold and Bu(), t £ M . Show that in this case estimate (4.8) has the form q()#()#, - ()m (), £ ()l; , q -q1 + A ps ps Cq M m + 1 . 176 Inertial Manifolds

C h § 5 Examples and Discussion a p t e r Example 5.1 3 Let us consider the nonlinear heat equation

ì ¶u ¶2 u ï = n + fx(),, u t,0< 0 , ()5.1 ï ¶t ¶x2 í u ==u 0 , () ï x = 0 xl= 5.2 ï u = u ()x . ()5.3 î t = 0 0 Assume that n is a positive parameter and fx(),, u t is a continuous function of its variables which possesses the properties M fx(),, u t - fx(),, u t £ Mu - u ,.fx(),,0 t £ 1 2 1 2 l Problem (5.1)–(5.3) generates a dynamical system in L2 ()0, l (see Section 3 of Chapter 2). Therewith 2 A = -n d ,,DA()= H1 ()0, L Ç H2 ()0, l dx2 0 where Hs()0, l is the Sobolev space of the order sB . The mapping (). , t given by the formula ux()® fx(),, ux()t satisfies conditions (1.2) and (1.3) with q = 0 . In this case spectral gap condition (2.3) has the form p2 n ()()N + 1 2 - N2 ³ 4M . l2 q Thus, problem (5.1)–(5.3) possesses an inertial manifold of the dimension N , provided that l2 N > - 1 + 2M (5.4) 2 n q p2 for some q < 22- .

Exercise 5.1 Find the conditions under which the inertial manifold of problem (5.1)–(5.3) is one-dimensional. What is the structure of the corresponding inertial form?

Exercise 5.2 Consider problem (5.1) and (5.3) with the Neumann boundary conditions: ¶u ¶u == 0 (5.5) ¶x ¶x x = 0 xl= Show that problem (5.1), (5.3), and (5.5) has an inertial manifold of the dimension N+ 1 , provided condition (5.4) holds for some Examples and Discussion 177

N ³ 0 . (Hint: A = -en()d2 ¤ dx2 + with condition (5.5), Bu()=, t =e- e u + fx(),, u t, where > 0 is small enough).

Exercise 5.3 Find the conditions on the parameters of problem (5.1), (5.3), and (5.5) under which there exists a one-dimensional inertial manifold. Show that if fx(),, u t º fu(), t , then the corresponding inertial form is of the type \ p ()t = fpt()(), t ,.p = p t = 0 0

Example 5.2 Consider the problem ì ¶ ¶2 ¶ u = n u + æö,,u , << > ï fxuèøt ,0 xl, t 0 , í ¶t ¶ x2 ¶x (5.6) ï u ==u 0, u = u ()x . î x = 0 xl= t = 0 0 Here n > 0 and fx(),,, u x t is a continuous function of its variables such that (),,,x - (),,,x £ - + x - x fx u1 t fx u2 t L1 u1 u2 L2 1 2 (5.7) for all x Î ()0, l , t ³ 0 and l [](),,, 2 £ 2 ò fx 00t dx L3 , 0 where Lj are nonnegative numbers. As in Example 5.1 we assume that 2 æö¶u A = -n d ,,DA()= H1 ()0, l Ç H2 ()0, l Bu(), t = fxu,, ,t . dx2 0 èø¶x It is evident that ¶u ¶u Bu(), t - Bu(), t £ L u - u + L 1 - 2 . 1 2 1 1 2 2 ¶x ¶x Here . is the norm in L2()0, l . By using the obvious inequality

¶ 2 æöp 2 u ³ u 2 ,,uHÎ 1 ()0, l ¶x èøl 0 we find that æöl Bu(), t - Bu(), t £ 1 L + L A12 ()u - u . 1 2 n èø1 p 2 1 2 Hence, conditions (1.2) and (1.3) are fulfilled with ìü q = 1 = ; 1 æöl + ,.MLmax íý3 èøL1 L2 2 îþn p 178 Inertial Manifolds

C h Therewith spectral gap condition (2.3) acquires the form a p 2 M p n ()2 N + 1 ³ ()2 N ++1 kN()+ 1 , t l q e r where 3 x 1 - -x p k == x 12e dx . 2 ò 2 0 Thus, the equation p + p q n 1 + × N 1 £ 2 2 N + 1 2 lM or p p p q n 2 ++ × 1 £ 2 2 2 N + 1 lM must be valid for some 0 <

Thus, in order to apply the above-presented theorems to the construction of the inertial manifold for problem (5.6) one should pose some additional conditions (see (5.7) and (5.9)) on the nonlinear term fx(),, u¶ u¤ ¶x ,t or require that the diffusion coefficient n be large enough.

Exercise 5.4 Assume that fx()e,,, u x t = fxu(),,,x t in (5.6), where the function f possesses properties (5.7) and (5.8) with arbitrary ³ Lj 0 . Show that problem (5.6) has an inertial manifold for any <

Exercise 5.5 Study the question on the existence of an inertial manifold for problem (5.6) in which the Dirichlet boundary condition is replaced by the Neumann boundary condition (5.5). Examples and Discussion 179

It should be noted that l = 2d ()+ () ® ¥ = W n Cd n 1 o 1 ,,n d dim , l where n are the eigenvalues of the linear part of the equation of the type ¶u = n Dufxuu+ (),,Ñ , t ,,,x Î W t > 0 ¶t in a multidimensional bounded domain W . Therefore, we can not expect that Theo- rem 3.1 is directly applicable in this case. In this connection we point out the paper [3] in which the existence of IM for the nonlinear heat equation is proved in a bounded domain W Ì Rd ()d £ 3 that satisfies the so-called “principle of spatial ave- raging” (the class of these domains contains two- and three-dimensional cubes). It is evident that the most severe constraint that essentially restricts an application of Theorem 3.1 is spectral gap condition (3.1). In some cases it is possible to weaken or modify it a little. In this connection we mention papers [6] and [7] in which spectral gap condition (3.1) is given with the parameters q = 2 and k = 0 for 0 £ q < 1 . Besides it is not necessary to assume that the spectrum of the operator AA is discrete. It is sufficient just to require that the selfadjoint operator possess a gap in the positive part of the spectrum such that for its edges the spectral condition holds. We can also assume the operator A to be sectorial rather than selfadjoint (for example, see [6]). Unfortunately, we cannot get rid of the spectral conditions in the construction of the inertial manifold. One of the approaches to overcome this difficulty runs as follows: let us consider the regularization of problem (0.1) of the form u d ++Au e Am u = Bu(), t,.(5.10)u = u dt t = 0 0

Here e > 0 and the number m > 0 is chosen such that the operator A = A + e Am possesses spectral gap condition (3.1). Therewith IM for problem (5.10) should be naturally called an approximate IM for system (0.1). Other approaches to the construction of the approximate IM are presented below.

It should also be noted that in spite of the arising difficulties the number of equations of mathematical physics for which it is possible to prove the existence of IM is large enough. Among these equations we can name the Cahn-Hillard equations in the domain W = ()0, L d , d =WdimW £ 2 , the Ginzburg-Landau equations (= ()0, L d , d £ 2 ), the Kuramoto-Sivashinsky equation, some equations of the theory of oscillations (d = 1 ), a number of reaction-diffusion equations, the Swift-Hohenberg equation, and a non-local version of the Burgers equation. The corresponding references and an extended list of equations can be found in survey [8]. In conclusion of this section we give one more interesting application of the theorem on the existence of an inertial manifold. 180 Inertial Manifolds

C h Example 5.3 a p Let us consider the system of reaction-diffusion equations t ¶ ¶ e u = n D + (), Ñ u = ¶ ufuu,,(5.11)¶ 0 r t n ¶W 3 W Ì d = (),,¼ (), x in a bounded domain R . Here uu1 um and the function fu satisfies the global Lipschitz condition: fu(), x - fv(), h £ Lu{}- v2 + xh- 2 12,(5.12) where uv, ÎxhRm , , ÎRmd , and L > 0 . We also assume that f ()00, £ L . Problem (5.11) can be rewritten in the form (0.1) in the space HL= []2 ()W m if we suppose Au = -n Duu+ ,.Bu()= ufuu+ (), Ñ It is clear that the operator A is positive in its natural domain and it has a discrete spectrum. Equation (5.12) implies that the relation ìü12 Bu()- Bv() £ Luíý- v2 + Ñ()uv- 2 + uv- £ îþ æöìüìü12 £ ç÷1 + Lmaxíý1 ; 1 íýuv- 2 + n Ñ()uv- 2 èøîþn îþ

is valid for Bu . Thus, Bu()- Bv() £ MA12 ()uv- , where ìü M = 1 + Lmaxíý1 ; 1 . îþn

Therefore, problem (5.11) generates an evolutionary semigroup St (see Chap- ()12 ter 2) in the space DA . An important property of St is the following: the subspace L which consists of constant vectors is invariant with respect to this semigroup. The dimension of this subspace is equal to m . The action of the semigroup in this subspace is generated by a system of ordinary differential equations u d = fu(), 0 ,.ut()Î L (5.13) dt

Exercise 5.6 Assume that equation (5.12) holds for xh==0 . Show that equation (5.13) is uniquely solvable on the whole time axis for any initial condition and the equation

ìü-Ls()- t sup íýe us() < ¥ (5.14) ts£ îþ holds for any s Î R . Examples and Discussion 181

The subspace L consists of the eigenvectors of the operator A corresponding to the l =l=mnm + eigenvalue 1 1 . The next eigenvalue has the form 2 1 1 , where 1 is the first nonzero eigenvalue of the Laplace operator with the Neumann boundary condition on ¶W . Therefore, spectral gap equation (3.1) can be rewritten in the form æöìü p nm ³ 2 ç÷+ ; 1 æöæö+ nm + + 1 1 Lmaxíý1 èøèø1 1 1 1 (5.15) q èøîþn 2 =q= ¤ <<- n > for Nm and 12 , where 0 q 22 . It is clear that there exists 0 0 nn³ such that equation (5.15) holds for all 0 . Therefore, we can apply Theorem 3.1 to find that if n is large enough, then there exists IM of the type ìü M = íýp + #()p : p Î#L, : L ® H L . îþ The invariance of the subspace L and estimate (5.14) enable us to use Theorem 3.2 and to state that LMÌ# . This easily implies that ()p º 0 , i.e. ML= . Thus, Theorem 3.1 gives us that for any solution ut() to problem (5.11) there exists a solution u ()t to the system of ordinary differential equations (5.13) such that

()- () £ -g t ³ ut u t 1 Ce ,,t 0 g > () . where the constant 0 does not depend on ut and 1 is the Sobolev norm of the first order.

Exercise 5.7 Consider the problem ¶u ¶2u = n + fx(), u,;0 <

n > 2 p()L2 + L2 , 521()- 1 2 182 Inertial Manifolds

C h then the dynamical system generated by problem (5.16) has the a two-dimensional (flat) inertial manifold p t Mp= {}sin x + p sin2 x : p , p Î e 1 2 1 2 R r and the corresponding inertial form is: 3 \ + n = (), \ + n = (), p1 p1 g1 p1 p2 ,.p2 4 p2 g2 p1 p2

Exercise 5.8 Study the question on the existence of an inertial manifold for the Hopf model of turbulence appearance (see Section 7 of Chap- ter 2).

§6§ 6Approximate Inertial Manifolds for Semilinear Parabolic Equations

Even in the cases when the existence of IM can be proved, the question concerning the effective use of the inertial form ¶ + = ()+ #(), , t pAp PB p pt t (6.1) is not simple. The fact is that it is not practically possible to find a more or less explicit solution to the integral equation for #()pt, even in the finite-dimensional case. In this connection we face the problem of approximate or asymptotic construction of an invariant (inertial) manifold. Various aspects of this problem related to finite-dimensional systems are presented in the book by Ya. Baris and O. Lykova [14]. For infinite-dimensional systems the problem of construction of an approximate IM can be interpreted as a problem of reduction, i.e. as a problem of constructive description of finite-dimensional projectors P and functions #(). , t : PH ® ® ()1 - P H such that an equation of form (6.1) “inherits” (of course, this needs to be specified) all the peculiarities of the long-time behaviour of the original system (0.1). It is clear that the manifolds arising in this case have to be close in some sense to the real IM (in fact, the dynamics on IM reproduces all the essential features of the qualitative behaviour of the original system). Under such a formulation a problem of construction of IM acquires secondary importance, so one can directly construct a sequence of approximate IMs. Usually (see the references in survey [8]) the problem of the construction of an approximate IM can be formulated as follows: find a surface of the form = {}+ #(), Î Mt p pt: pPH,(6.2) which attracts all the trajectories of the system in its small vicinity. The character of l-1 closeness is determined by the parameter N +1 related to the decomposition Approximate Inertial Manifolds for Semilinear Parabolic Equations 183

dp ì + Ap = PB() p+ q, t , ï dt í (6.3) ï dq î + Aq = ()1 - P Bp()+ q, t . dt ()0 #()#, = (), º We obtain the trivial approximate IM Mt if we put pt 0 pt 0 in (6.2). ()0 0 In this case Mt is a finite-dimensional subspace in whereas inertial form (6.1) turns into the standard Galerkin approximation of problem (0.1) corresponding to ()1 this subspace. One can find the simplest non-trivial approximation Mt using formula (6.2) and assuming that #()#, = (), º -1()- (), pt 1 pt A 1 P Bp t.(6.4) ()1 The consideration of system (0.1) on Mt leads to the second equation of equations (6.3) being replaced by the equality A q = ()1 - P Bp(), t . The results of the computer simulation (see the references in survey [8]) show that the use of just the first approximation to IM has a number of advantages in comparison with the traditional Galerkin method (some peculiarities of the qualitative behaviour of the system can be observed for a smaller number of modes).

There exist several methods of the construction of an approximate IM. We present the approach based on Lemma 2.1 which enables us to construct an approximate IM of the exponential order, i.e. the surfaces in the phase space H such that their expo- l nentially small (with respect to the parameter N +1 ) vicinities uniformly attract all the trajectories of the system. For the first time this approach was used in paper [15] for a class of stochastic equations in the Hilbert space. Here we give its deterministic version. Let us consider the integral equation (see(2.1)) ()= sL, []() - ££ vt Bp v t , sL ts =rrl-q and assume that L N +1 , where the parameter possesses the property æöqq -()- q qMº l 1 r1- q + r < 1 .(6.5) èø1 - q 1 In this case equations (2.2) hold. Hence, Lemma 2.1 enables us to construct a collec- {}L = rl-q tion of manifolds Ms for L N +1 with the help of the formula L = {}+ #L(), Î Ms p ps: pPH,(6.6) where s -()- t #L()ps, = ò e s A QB()t v()tt , d º Qv() s, p .(6.7) sL- ()= (), = rl-q Here vt vt p is a solution to integral equation (2.1) and L N +1 . 184 Inertial Manifolds

C #L, L h Exercise 6.1 Show that both the function ps and the surface Ms a (), º p do not depend on sBut in the autonomous case Bt . t e r The following assertion is valid.

3 Theorem 6.1. r =LLr (),,ql = (),,ql There exist positive numbers 1 1 M 1 and M 1 such that if -q l1- q ³ Lr-1 = rl < rr£ N +1 ,,,L N +1 ,,,0 1 , (6.8) then the mappings #L(). , s : PH® QH defined by equation (6.7) possess the property Aq()Qu() t - #L()Pu() t , t £

ìüs ìür ()1 0 q ()2 1- q £ C expíý- r l + ()tt- + C exp íý- l + (6.9) R îþN 1 * R îþ2 N 1 ³s+ ¤ > () for all tt* L 2 . Here 0 0 is an absolute constant and ut is a mild solution to problem (1.1) such that q () £ Î [ ,)+¥ A ut R for tt* . (6.10) If Bu(), t £ M , then estimate (6.9) can be rewritten as follows:

Aq()Qu() t - #L()Pu() t , t £

ìüs 0 q 1- q £ C exp íý- r l + ()tt- + D exp {}-rl + , (6.11) R îþN 1 * 2 N 1

where D2 is defined by equality (2.14).

Proof. Let ()= (), ; ()+ #L()(), ££ u t Ut s Pus Pu s s ,,t* st where Ut(), s; v is a mild solution to problem (1.1) with the initial condition Î ()q ()= (), ; vDA at the moment sut . Therewith Ut 0 u0 . It is evident that Qu() t - #L()Pu() t , t = Qut()()- u ()t +

++Qu ()t -# #L()Pu ()t , t L()#Pu ()t , t - L()Pu() t , t . (6.12)

Let us estimate each term in this decomposition. Equation (1.6) implies that q ()a()- () £ ()- q()()- #L()(), A Qut u t N tsA Qu s Pu s s ,(6.13) Approximate Inertial Manifolds for Semilinear Parabolic Equations 185 where -l t a t a ()t = N +1 + ()+ l-q1 + 2 N e M 1 k a1 N +1 e . Using (2.16) we find that q()b ()- #L() (), £ (), - A Qu s Pu s s N Lt s (6.14) where b (), t = ()- -1 -g L + ()- -2 ()+ -gt N L D2 1 q e q 1 q RD1 e , b (), t (), £ moreover, the second term in N L can be omitted if Bu t M (see Exer- cise 2.1). At last equations (2.15) and (1.5) imply that

Aq()##L()Pu ()t , t - L()Pu() t , t £

q a ()ts- £ a e 2 Aq()Qu() s - #L()Pu() s , s . (6.15) 1 1- q Thus, equations (6.12)–(6.15) give us the inequality ()£ a ()- ()+ b (), - dt N tsds N Lt s (6.16) ³³ for tst* , where dt()= Aq()Qu() t - #L()Pu() t , t and

q a t a ()t == a ()t + a e 2 N N 1 1- q -l t a t = N +1 + ()l+ -q1 + + ()- -1 2 e a1 M 1 k N +1 q 1 q e .

It follows from (6.16) that under the condition sL+ ¤ 2 ££tsL+ the equation ()£ a ()+ b (), ¤ dt NL, ds N LL2 (6.17) holds with L -l -q+ q a L a = e N + 1 2 + a M ()l1+ k 1 + e 2 . NL, 1 N +1 1- q a £ ¤ It is clear that NL, 12 if l >l-q1 + > ()+ N +1 L 42ln , N +1 16 a1M 1 k and < £ ()+ -1 a2 L ln2 ,.(6.18)q 116a1 r = r (),,ql = rl-q Let 1 1 M 1 be such that equation (6.18) holds for L N +1 and for < rr£ the parameter q of the form (6.5) with 0 1 . Then equation (6.8) with L =a()+ ()r+ £ ¤ = + ()¤ 41 4a1 M 1 k 1 implies that NL, 12 . Let tn t* 12nL . Then it follows from (6.17) that 186 Inertial Manifolds

C h 1 dt()+ £ dt()+ b()LL, ¤ 2 , n = 012,,,¼ a n 1 2 n N p t After iterations we find that e r ()£ -n ()+ b æö, L = ,,,¼ dtn 2 dt0 2 N èøL ,(6.19)n 012 3 2 Equation (6.17) also gives us that æö dt()£ 1 dt()+ b L, L ,.t + L ££tt+ L 2 n N èø2 n 2 n Therefore, it follows from (6.19) that ìü () £ -2 ()- ()+ b æö, L dt 2 expíýtt* ln2 dt* 2 N èøL îþL 2 ³gl+ ¤ = for all tt* L 2 . This implies (6.9) and (6.11) if we take N +1 in the equa- b (), ¤ tion for N LL2 . Thus, Theorem 6.1 is proved.

In particular, it should be noted that relations (6.9) and (6.11) also mean that a solution to problem (0.1) possessing the property (6.10) reaches the layer of the thick- e = {}-l1- q {}L ness N c1 exp c2 N +1 adjacent to the surface Mt given by equation (6.6) for t large enough. Moreover, it is clear that if problem (0.1) is autonomous ()Bu, t º Bu and if it possesses a global attractor, then the attractor lies in this layer. In the autonomous case ML does not depend on t (see Exercise 6.1). These observations give us some information about the position of the attractor in the phase space. Sometimes they enable us to establish the so-called localization theorems for the global attractor.

Exercise 6.2 Let Bu(), t £ M . Use equations (1.4) and (1.8) to show that -l ()ts- q () £ 1 q + A ut e A u0 R0 , = ()l+ -1 + q where R0 M 1 k 1 .

In particular, the result of this exercise means that assumption (6.10) holds for any > (), £ RR0 and for t* large enough under the condition Bu t M . In the general case equation (6.10) is a variant of the dissipativity property.

= L (), ()- , Exercise 6.3 Let v0 v0, s ts be a function from Cgq, sLs . Assume that L º L (), = sL, []() = ,,¼ vn vns, ts Bp vn - 1 t , n 12 and #L(), = L (), = ,,,¼ n ps Qvns, sp, n 012 Show that the assertions of Theorem 6.1 remain true for the function #L(), n ps if we add the term Approximate Inertial Manifolds for Semilinear Parabolic Equations 187

qn()v + c ()D + R 0 s 0 1 to the right-hand sides of equations (6.9) and (6.11). Here q is defined by equality (6.5) and v is the norm of the function v in the 0 s 0 space Cgq, ()sL- , s .

#L(), Therefore, the function n ps generates a collection of approximate inertial l manifolds of the exponential (with respect to N +1 ) order for n large enough.

Example 6.1 Let us consider the nonlinear heat equation in a bounded domain W Ì Rd : ì ¶u ï = Dufu+ (), Ñu , x Î W , ts> , í ¶t (6.20) ï u ==0, u u ()x . î ¶W ts= 0

Assume that the function fw(), x possesses the properties (), x - (), x £ ()-x+ - x (), x £ fu1 1 fu2 2 C1 u1 u2 1 2 , fu C2 . We use Theorem 6.1 and the asymptotic formula l ~ 2d ® ¥ N c0 N ,,N for the eigenvalues of the operator -WD in Ì Rd to obtain that in the Sobolev 1()W space H0 for any N there exists a finite-dimensional Lipschitzian surface MN of the dimension N such that ()(), £ {}-s 1d ()- + {}-s 1d dist 1()W ut MN C1exp 1 N tt* C2 exp 2 N H0 ³ 1()W () for tt* and for any mild (in H0 ) solution ut to problem (6.20). Here t* s is large enough, Cj and j are positive constants.

Exercise 6.4 Consider the abstract form of the two-dimensional system of the Navier-Stokes equations u d ++nAu b() u, u = ft(),(6.21)u = u dt t = 0 0 (see Example 3.5 and Exercises 4.10 and 4.11 of Chapter 2). Assume that A12 ft() < C for t ³ 0 . Use the dissipativity property for (6.21) and the formula l c k ££k c k 0 l 1 1 for the eigenvalues of the operator A to show that there exists a collection of functions {}#()pt, : t ³ 1 from PD() A into ()1-P DA() possessing the properties 188 Inertial Manifolds

C #(), £ -12 h a)A pt c1 N ; a ()#()#, - (), £ ()- p A p1 t p2 t c2 Ap1 p2 t , , Î () e for any pp1 p2 PD A ; r b) for any solution ut()Î DA() to problem (6.21) there 3 exists t* > 1 such that AQ() u() t - #()Pu() t , t £

£ {}-s 12 ()- + {}-s 12 c3 exp 0 N tt* c4 exp 1 N . Here PN is the orthoprojector onto the first eigenelements of the operator A .

Exercise 6.5 Use Theorem 6.1 to construct approximate inertial manifolds for (a) the nonlocal Burgers equation, (b) the Cahn-Hilliard equation, and (c) the system of reaction-diffusion equations (see Sec- tions 3 and 4 of Chapter 2).

In conclusion of the section we note (see [8], [9]) that in the autonomous case the approximate IM can also be built using the equation áñ#¢()p ; - Ap + PB() p + #()p + A#()p = QB() p + #()p .(6.22) Here pPHÎ , QIP=#¢- , ()p is the Frechét derivative and áñ #¢()p , w is its value at the point pw on the element . At least formally, equation (6.22) can be obtained if we substitute the pair {}pt(); #()pt() into equation (6.3). The second of l-1 equations (6.3) implicitly contains a small parameter N +1 . Therefore, using (6.22) {}# we can suggest an iteration process of calculation of the sequence m giving the approximate IM:

A# ()p = QB() p+ #n ()()p + k 1 k ¢ + áñ#n ()()p ; Ap- PB() p + #n ()k ()p ,,(6.23)k ³ 1 2 k 3 n () where the integers i k are such that ££n () - n()= ¥ = ,, 0 i k k 1 ,,.lim i k i 123 k ® ¥ One should also choose the zeroth approximation and concretely define the form of n () #()ºn()= - = ,, the values i k (for example, we can take 0 v 0 and i k k 1 , i 12 3 ). When constructing a sequence of approximate IMs one has to solve only a linear stationary problem on each step. From the point of view of concrete calculations this gives certain advantages in comparison with the construction used in Theorem 6.1. However, these manifolds have the power order of approximation only (for detailed discussion of this construction and for proofs see [9]).

# () Exercise 6.6 Prove that the mapping 1 v has the form (6.4) under the # ()º#() condition 0 v 0 . Write down the equation for 2 v when n ()=n()== n() 1 2 1 ,.2 2 3 2 0 Inertial Manifold for Second Order in Time Equations 189

§ 7 Inertial Manifold for Second Order in Time Equations

The approach to the construction of IM given in Sections 2–4 is essentially based on the fact that the system has form (0.1) with a selfadjoint positive operator A . How- ever, there exists a wide class of problems which cannot be reduced to this form. From the point of view of applications the important representatives of this class are second order in time systems arising in the theory of nonlinear oscillations:

ì 2 u u ï d ++2 e d Au = Bu(), t, ts> , e > 0, ï dt2 dt í (7.1) ï u u ==u , d u . ï ts= 0 dt 1 î ts= In this section we study the existence of IM for problem (7.1). We assume that m A is a selfadjoint positive operator with discrete spectrum (k and ek are the corresponding eigenvalues and eigenelements) and the mapping Bu(), t possesses the properties of the type (1.2) and (1.3) for 0 ££q 12¤ , i.e. Bu(), t is a continuous mapping from DA()q ´ R into H such that (), £ B 0 t M0 , (), - (), £ q()- Bu1 t Bu2 t M1 A u1 u2 ,(7.2) ££q ¤ , Î ())q º where 0 12 and u1 u2 D q .

The simplest example of a system of the form (7.1) is the following nonlinear wave equation with dissipation:

ì ¶2u ¶u ¶2u æö¶u ï + 2 e - + fxt, ; u, = 0, 0 < , ï ¶t2 ¶t ¶x2 èø¶x ï == í u = u = 0 , (7.3) ï x 0 xL ï ¶u u ==u ()x , u ()x . ï ts= 0 ¶t 1 î ts=

Let 0 = DA()12 ´ H . It is clear that 0 is a separable Hilbert space with the inner product (), = (), + (), UV Au0 v0 u1 v1 ,(7.4) = (); = (); 00 where Uu0 u1 and Vv0 v1 are elements of . In the space problem (7.1) can be rewritten as a system of the first order: d Ut()+ AUt()= B()Ut(), t ,;ts> U = U . (7.5) dt ts= 0 190 Inertial Manifolds

C h Here a æö¶u()t p Ut()= ut(); ,.U = ()u ; u Î 0 t èø¶t 0 0 1 e r The linear operator AB and the mapping ()Ut, are defined by the equations: 3 = ()- ; + e ()= ()´ ()12 AUu1 Au0 2 u1 ,,(7.6)D A DA DA (), = (); (), = (); B Ut 0 Bu0 t ,.Uu0 u1

Exercise 7.1 Prove that the eigenvalues and eigenvectors of the operator A have the form: l± = ee± 2 - m ± = (); -l± = ,,¼ n n ,,,(7.7)fn en n en n 12 m where n and en are the eigenvalues and eigenvectors of A . Exercise 7.2 Display graphically the spectrum of the operator A on the complex plane.

These exercises show that although problem (7.1) can be represented in the form (7.5) which is formally identical to (0.1) we cannot use Theorem 3.1 here. Neverthe- less, after a small modification the reasoning of Sections 2–4 enables us to prove the existence of IM for problem (7.1). Such a modification based on an idea from [16] is given below. First of all we prove the solvability of problem (7.1). Let us first consider the linear problem

ì 2u u ï d ++2 e d Au = ht(), ts> , ï dt2 dt í (7.8) u ï u ==u , d u . ï ts= 0 dt 1 î ts=

These equations can also be rewritten in the form (cf. (7.5)) d Ut()+ AUt()= Ht(),,U = U (7.9) dt ts= 0 \ where Ut()= ()ut(); u()t and Ht()= ()0 ; ht() . We define a mild solution to problem (7.8) (or (7.9)) on the segment []ss, + T as a function ut() from the class º (), + ; ÇÇ1 (), + ; 2 (), + ; LsT, Cs s T 12¤ C ss TH C ss T -1¤ 2 q which satisfies equations (7.8). Here q = DA() as before (see Chapter 2). One can prove the existence and uniqueness of mild solutions to (7.8) using the Galerkin method, for example. The approximate Galerkin solution of the order m is defined as a function Inertial Manifold for Second Order in Time Equations 191

m ()= () um t å gk t ek k = 1 satisfying the equations

ì ()\\ (), ++e ()\ (), ()(), = ()(), > ï um t ej 2 um t ej Aum t ej ht ej , ts , í (7.10) ï ()(), ==(), ()\ (), (), î um s ej u0 ej , um s ej u1 ej \ for j = 12,,¼ ,m . Moreover, we assume that g ()t Î C1()ss, + T and g ()t is j \ j absolutely continuous. Hereinafter we use the notation v ()t = dv¤ dt . Evidently equations (7.10) can be rewritten in the form ì \\ ()++e \ () ()= () ï um t 2 um t Aum t pmht , í (7.11) = \ = ï um pm u0 , um pmu1 , î ts= ts= {}, ¼ where pm is the orthoprojector onto Lin e1 em in H .

In the exercises given below it is assumed that ()Î ¥(), Î ()12 Î ht L R H ,,.(7.12)u0 DA u1 H

Exercise 7.3 Show that problem (7.10) is uniquely solvable on any segment [], + ()Î ss T and um t LsT, . Exercise 7.4 Show that the energy equality t 1 æö\ \ u ()t 2 + A12 u ()t 2 + 2 e u ()t 2 dt = 2 èøm m ò m s t 1 æö2 + 12 2 \ = p u A pmu + ()th ()t , u ()t d (7.13) 2 èøm 1 0 ò m s holds for any solution to problem (7.10).

Exercise 7.5 Using (7.11) and (7.13) prove the a priori estimate -12 \\ ()2 ++\ ()2 12 () £ (),, A um t um t A um t CT u0 u1 for the approximate Galerkin solution um t to problem (7.8). Exercise 7.6 Using the linearity of problem (7.11) show that for every two () () approximate solutions um t and um¢ t the estimate -1¤ 2 ()\\ ()- \\ () 2 + A um t um¢ t

++\ ()- \ () 12 ()()- () 2 £ um t um¢ t A um t um¢ t 192 Inertial Manifolds

C h £ C ()p - p u 2 + a T m m¢ 1 p t ++A12 ()p - p u 2 ess sup ()p - p h ()t 2 e m m¢ 0 m m¢ t Î []ss, + T r Î [], + > 3 holds for all tssT , where T 0 is an arbitrary number. Exercise 7.7 Using the results of Exercises 7.5 and 7.6 show that we can pass to the limit n ® ¥ in equations (7.11) and prove the existence and uniqueness of mild solutions to problem (7.8) on every segment []ss, + T under the condition (7.12).

Exercise 7.8 For a mild solution ut() to problem (7.8) prove the energy equation: t 1 æö\ \ u()t 2 + A12 ut()2 + 2 e u()t 2 dt = 2 èøò s t 1 æö\ = u 2 + A12 u 2 + ()th ()t , u()t d . (7.14) 2 èø1 0 ò s

In particular, the exercises above show that for ht()º 0 problem (7.8) generates a linear evolutionary semigroup e-t A in the space 0 = DA()12 ´ H by the formula -t A (); = ()(); \ () e u0 u1 ut u t ,(7.15) where ut() is a mild solution to problem (7.8) for ht()º 0 . Equation (7.14) implies that the semigroup e-t A is contractive for e ³ 0 .

Exercise 7.9 Assume that conditions (7.12) are fulfilled. Show that the mild solution to problem (7.8) can be presented in the form t ()\ (); () = -()ts- A (); + -()ts- A ()t; ()t u t ut e u0 u1 òe 0 h d ,(7.16) s where the semigroup e-t A is defined by equation (7.15).

Let us now consider nonlinear problem (7.1) and define its mild solution as \ a function Ut()º ()ut(); u()t Î Cs(), s+ T; 0 satisfying the integral equation t ()= -()ts- A + -()ts- A ()t()t , t Ut e U0 òe B U d (7.17) s [], + ()(), = (); ()(), = (); on ss T . Here B Ut t 0 But t and U0 u0 u1 . Inertial Manifold for Second Order in Time Equations 193

Exercise 7.10 Show that the estimates () B()Ut, 0 £ M 1 + U 0 , (), - (), £ - B U1 t B U2 t 0 MU1 U2 0 hold in the space 0 = DA()12 ´ H . Here M is a positive constant.

Exercise 7.11 Follow the reasoning used in the proof of Theorems 2.1 and 2.3 of Chapter 2 to prove the existence and uniqueness of a mild solution to problem (7.1) on any segment []ssT, + .

Thus, in the space 0 there exists a continuous evolutionary family of operators St(), s possessing the properties St(), t = I ,,St(), t ° S ()t, s = St(), s and (), = ()(); \ () St sU0 ut u t , () = where ut is a mild solution to problem (7.1) with the initial condition U0 = (); u0 u1 . e2 > m Let condition N +1 hold for some integer N . We consider the decomposition of the space 0 into the orthogonal sum 00= Å 0 1 2 , where 0 = {}(); , (); = ,,¼ 1 Lin ek 0 0 ek : k 12 N 0 and 2 is defined as the closure of the set {}(); , (); ³ + Lin ek 0 0 ek : kN1 .

0 0 Exercise 7.12 Show that the subspaces 1 and 2 are invariant with respect to the operator A . Find the spectrum of the restrictions of the operator A to each of these spaces.

0 0 Let us introduce the following inner products in the spaces 1 and 2 (the purpose of this introduction will become apparent further): áñUV, = e2 ()u , v - ()eAu , v + ()u +eu , v + v , 1 0 0 0 0 0 1 0 1 (7.18) áñ, = ()e, ++()2 - m ()e, ()+e, + UV2 Au0 v0 2 N +1 u0 v0 u0 u1 v0 v1 . = (); = (); Here Uu0 u1 and Vv0 v1 are elements from the corresponding sub- 0 0 space i . Using (7.18) we define a new inner product and a norm in by the equalities: áñUV, = áñU , V + áñU , V ,,UUU= áñ, 12 1 1 1 2 2 2 = + = + where UU1 U2 and VV1 V2 are decompositions of the elements UV and , Î 0 = , into the orthogonal terms Vi Ui i ,i 12 . 194 Inertial Manifolds

C h Lemma 7.1. a p The estimates t ³ 1 e2 - m q = (); Î 0 e U 1 q N A u0 ,;Uu0 u1 1 (7.19) r m N 3 U ³ 1 d Aqu , Uu= (); u Î 0 (7.20) 2 mq N, e 0 0 1 2 N + 1 hold for 0 ££q 12¤ . Here æöe2 - m d = m ç÷, N +1 N, e N +1 min 1 m . (7.21) èøN +1

Proof. = (); Î 0 b £ mb Let Uu0 u1 1 . It is evident that in this case A u0 N u0 for any b > 0 . Therefore, 2 ³³e2 2 -m12 2 -2 q ()e2 - m q 2 U 1 u0 A u0 N N A u0 , Î 0 i.e. equation (7.19) holds. Let U 2 . Then using the inequality b ³ mb b > Î {}³ + A u0 N +1 u0 ,,0 u0 Lin ek : kN1 (7.22) for 0 < d £ 1 we find that 2 ³ d2 12 2 + ()e2 - ()m+ d2 2 U 2 A u0 1 N +1 u0 . dd= m-12 If we take N, e N +1 and use (7.22), then we obtain estimate (7.20). The lemma is proved.

In particular, this lemma implies the estimate q £ mq d-1 A u0 N +1 N, e U (7.23) = (); Î 0 ££q ¤ d for any Uu0 u1 , where 0 12 and N, e has the form (7.21).

Exercise 7.13 Prove the equivalence of the norm . and the norm generated by the inner product (7.4). d = e2 - m q = Exercise 7.14 Show that we can take N, e N +1 for 0 in (7.20) and (7.23). {}± Exercise 7.15 Prove that the eigenvectors fk of the operator A (see (7.7)) possess the following orthogonal properties: áñ+, + ==áñ–, – áñ+, – =¹ fn fk fn fk fn fk 0 ,,kn áñ+, – = ££ fk fk 0 ,.(7.24)1 kN Inertial Manifold for Second Order in Time Equations 195

Note that the last of these equations is one of the reasons of introducing a new inner product.

Let P0 be the orthoprojector onto the subspace 0 in 0 , i = 12, . i i Lemma 7.2. The equality -l- -At N +1 t e P0 = e ,,t ³ 0 (7.25) 2 is valid. Here . is the operator norm which is induced by the corresponding vector norm.

Proof. Îy0 ()= -At 2 0 Let U 2 . We consider the function t e U . Since 2 is invariant with respect to e-A t , the equation y()= ()e(), () ++()2 - m ()(), () \ + e 2 t Au t ut 2 N +1 ut ut u u holds, where ut() is a solution to problem (7.8) for ht()º 0 . After simple calculations we obtain that dy \ + 2 ey = 4 ()e2 - m ()u + e u, u . dt N +1 It is evident that

e2 - m ()e\ + e , ££()2 - m 2 +y\ + e 2 () 2 N +1 u u u N +1 u u u t . Therefore, dy + 2 ey £ 2 e2 -ym . dt N +1 Consequently, - -l t y()t £ e N +1 y()0 ,.t > 0 (7.26) If we now notice that - - -l t - {}- = N +1 exp At fN +1 e fN +1 , then equation (7.26) implies (7.25). Thus, Lemma 7.2 is proved.

Let us consider the subspaces ± 0 ± = {}£ 1 Lin fk : kN. Equation (7.24) gives us that the subspaces are orthogonal to each other and there- 0 = 0 +Å 0 – fore 1 1 1 . Using (7.24) it is easy to prove (do it yourself) that 196 Inertial Manifolds

C - h l t At - £ N Î a e P0 e ,,t R (7.27) p 1 t e + - -l t r At + £ N ,.> (7.28) e P0 e t 0 1 3 We use the following pair of orthogonal (with respect to the inner product áñ.., ) projectors in the space 0

= - , ==- + PP0 QIPP0 + P0 1 1 2 to construct the inertial manifold of problem (7.1) (or (7.5)). Lemma 7.2 and equations (7.27) and (7.28) imply the dichotomy equations - - l t -l t eAtPe£ N ,;t Î R e-At Qe£ N +1 ,.(7.29)t > 0 l- = ee- 2 - m e2 > m We remind that N k and N +1 . The assertion below plays an important role in the estimates to follow.

Lemma 7.3. (), = (); (), = (); Î 0 () Let B Ut 0 Bu0 t , where Uu0 u1 and Bu0 possesses properties (7.2). Then (), £ + Î 0 B Ut M0 KN U ,,U (), - (), £ - , Î 0 B U1 t B U2 t KN U1 U2 ,,(7.30)U1 U2 where

q - ¤ æöm + K = M m 12max ç÷1, N 1 . (7.31) N 1 N +1 e2 - m èøN +1

The proof of this lemma follows from the structure of the mapping B()Ut, and from estimates (7.2) and (7.23).

=q()e2 - m -12 = Exercise 7.16 Show that one can take KN M1 N +1 for 0 in (7.30) (Hint: see Exercise 7.14).

Let us now consider the integral equation (cf. (2.1) for L = ¥ ) ()= Bs []()º Vt p V t

s t º e-()ts- Ape- ò -()t - t A PB()tV()tt , d + ò e-()t - t A QB()tV()tt , d (7.32) t -¥ Inertial Manifold for Second Order in Time Equations 197

()(- ¥, ] in the space Cs of continuous vector-functions Ut on s with the values in0 such that the norm I I º -g ()st- () < ¥ g = 1 ()l- + l- U sup e Ut ,,N +1 N ts< 2 is finite. Here pPÎ 0 andt Î ()-¥, s .

Exercise 7.17 Show that the right-hand side of equation (7.32) is a continuous function of the variable t with the values in 0 .

Lemma 7.4. Bs The operator p maps the space Cs into itself and possesses the properties

æö1 1 4 K I Bs []V I £ pM++ç÷ + N I VI (7.33) p 0 èøl- l- l- - l- N N +1 N +1 N and 4 K I Bs []V - Bs []V I £ N IV - V I . (7.34) p 1 p 2 l- - l- 1 2 N +1 N

Proof. Let us prove (7.34). Evidently, equations (7.29) and (7.30) imply that

s - l ()t - t Bs []() - Bs []() £ N ()t -t()t + p V1 t p V2 t KN òe V1 V2 d t t - -l ()t - t + N +1 ()t -t()t KN ò e V1 V2 d . -¥ Since ()t - ()t £ g ()s - t I - I V1 V2 e V1 V2 , it is evident that Bs []()- Bs []() £ g ()st- I - I p V1 t p V2 t qe V1 V2 with ìü s - t - ïï()tl - g ()- t -()l - g ()t - t qK= íýe N dt + e N +1 dt . N ïïò ò îþt -¥ £ ()l- - l- -1 Simple calculations show that q 4 KN N +1 N . Consequently, equation (7.34) holds. Equation (7.33) can be proved similarly. Lemma 7.4 is proved. 198 Inertial Manifolds

C < h Thus, if for some q 1 the condition a - - 4K p l - l ³ N (7.35) t N +1 N q e r holds, then equation (7.32) is uniquely solvable in Cs and its solution V can be esti- 3 mated as follows: - æöæö1 1 IV I £ ()1 - q 1 pM+ + .(7.36) èø0 èøl- l- N N +1 {} 0 Therefore, we can define a collection of manifolds Ms in the space by the formula = {}+ #(), Î 0 Ms p ps: pP ,(7.37) where s #()ps, = ò e-()t - t A QB()tV()tt , d .(7.38) -¥ Here V()t is a solution to integral equation (7.32). The main result of this section is the following assertion.

Theorem 7.1. Assume that 4K e2 >lm - - l- ³ N N +1 and N +1 N q (7.39) << l- = ee- 2 - m for some 0 q 1 , where k k and KN is defined by formula (7.31). Then the function #()ps, given by equality (7.38) satisfies the Lip- schitz condition q #()#p , s - ()p , s £ p - p (7.40) 1 2 21()- q 1 2

and the manifold Ms is invariant with respect to the evolutionary operator St(), t generated by the formula (), t = ()(); \ () ³ St U0 ut u t ,,,ts, in 0 , where ut() is a solution to problem (7.1) with the initial condition = (); <<- U0 u0 u1 . Moreover, if 0 q 22, then there exist initial conditions * = ()* ; * Î U0 u0 u1 Ms such that (), - (), * £ -g ()ts- - #(), St sU0 St sU0 Cq e QU0 PU0 s 1 - - for ts³g , where = ()l + l . 2 N N +1 The proof of the theorem is based on Lemma 7.4 and estimates (7.29) and (7.30). It almost entirely repeats the corresponding reasonings in Sections 2 and 3. We give the reader an oppotunity to recover the details of the reasonings as an exercise. Inertial Manifold for Second Order in Time Equations 199

Let us analyse condition (7.39). Equation (7.31) implies that (7.39) holds if m - m e2 ³ m N +1 N ³ 4 mq - 12¤ 2 N +1 ,.(7.41)M1 N +1 e2 - m q 2 N However, if we assume that

82 q m - m ³ M m ,(7.42) N +1 N q 1 N +1 then for condition (7.41) to be fulfilled it is sufficient to require that m ££e2 m + m 2 N +1 2 N +1 N .(7.43) Thus, if for some N conditions (7.42) and (7.43) hold, then the assertions of Theo- rem 7.1 are valid for system (7.1). This enables us to formulate the assertion on the existence of IM as follows.

Theorem 7.2. m Assume that the eigenvalues N of the operator A possess the properties m N >m=rr ()+ () > ® ¥ infm 0 and Nk()+1 c0 k 1 o 1 ,,, 0 ,,,k , (7.44) N N +1 for some sequence {}Nk() which tends to infinity and satisfies the estimate

m - m ³ 82 mq <<- Nk()+1 Nk() q M1 Nk()+1 ,,.0 q 22. e > Then there exists 0 0 such that the assertions of Theorem 7.1 hold for all ee³ 0 .

Proof.

Equation (7.44) implies that there exists k0 such that the intervals []m , m + m ³ 2 Nk()+1 2 Nk()+1 Nk(),,kk0 [e ,)+¥ cover some semiaxis 0 . Indeed, otherwise there would appear a subse- {}() quence Nkj such that m ()< 2 ()m ()+ + - m ()+ Nkj Nkj 1 1 Nkj 1 ee³ But that is impossible due to (7.44). Consequently, for any 0 there exists NN= e such that equations (7.42), (7.43) as well as (7.39) hold. ¶ Exercise 7.18 Consider problem (7.3) with the function fx()=,, t u ,u ¶x = fx(),, t u possessing the property (),, - (),, £ - fx t u1 fx t u2 Lu1 u2 . Use Theorem 7.1 to find a domain in the plane of the parameters ()e, L for which one can guarantee the existence of an inertial manifold. 200 Inertial Manifolds

C h § 8 Approximate Inertial Manifolds a p for Second Order in Time Equations t e r 3 As seen from the results of Section 7, in order to guarantee the existence of IM for aproblem of the type 2u u d ++g d Au = Bu() , dt2 dt (8.1) u u ==u , d u , t = 0 0 t 1 d t = 0 we have to require that the parameter g = 2 e > 0 be large enough and the spectral gap condition (see (7.41)) be valid for the operator A . Therefore, as in the case with parabolic equations there arises a problem of construction of an approximate inertial manifold without any assumptions on the behaviour of the spectrum of the operator A and the parameter g > 0 which characterizes the resistance force. Unfortunately, the approach presented in Section 6 is not applicable to the equation of the type (8.1) without any additional assumptions on g . First of all, it is connected with the fact that the regularizing effect which takes place in the case of parabolic equations does not hold for second order equations of the type (8.1) (in the parabolic case the solution at the moment t > 0 is smoother than its initial condition). In this section (see also [17]) we suggest an iteration scheme that enables us to construct an approximate IM as a solution to a class of linear problems. For the sake of simplicity, we restrict ourselves to the case of autonomous equations (Bu, t º º)Bu. The suggested scheme is based on the equation in functional derivatives such that the function giving the original true IM should satisfy it. This approach was developed for the parabolic equation in [9] (see also [8]). Unfortunately, this approach has two defects. First, approximate IMs have the power order (not the exponential one as in Section 6) and, second, we cannot prove the convergence of approximate IMs to the exact one when the latter exists.

Thus, in a separable Hilbert space H we consider a differential equation of the type (8.1) where g is a positive number, A is a positive selfadjoint operator with discrete spectrum and B (). is a nonlinear mapping from the domain DA()12 of the operator A12 into Hm such that for some integer ³ 2 the function Bu() lies in Cm as a mapping from DA()12 into H and for every r > 0 the following estimates hold:

k áñB()k ()u ; w ,,¼ w £ C A12 w ,(8.2) 1 k r Õ j j = 1 Approximate Inertial Manifolds for Second Order in Time Equations 201

k áñ()k ()- ()k ()* ; ,,¼ £ 12 ()- * 12 B u B u w1 wk Cr A uu Õ A wj ,(8.3) j = 1 where k = 01,,¼ ,m , . is a norm in the space HA , 12 u £ r , A12 u* £ r , Î ()12 ()k () () and wj DA . Here B u is the Frechét derivative of the order kBu of áñ()k (); ,,¼ ,,¼ and B u w1 wk is its value on the elements w1 wk . Let LmR, be a class of solutions to problem (8.1) possessing the following properties of regularity: I) for k = 01,,¼ ,m - 1 and for all T > 0 u()k ()t Î C()0, TDA; () and u()m ()t Î C ()0, TDA; ()12 ,,u()m + 1 ()t Î C ()0, TH; where C ()0, TV; is the space of strongly continuous functions on []0, T ()k ()= ¶k () with the values in Vu , hereinafter t t ut ; Î II) for any uLmR, the estimate u()k + 1 ()t 2 ++A12 u()k ()t 2 Au()k - 1 ()t 2 £ R2 (8.4) = ,,¼ ³ * * holds for k 1 m and for tt , where t depends on u0 and u1 only. In fact, the classes LmR, are studied in [18]. This paper contains necessary and sufficient conditions which guarantee that a solution belongs to a class LmR, . It should be noted that in [18] the nonlinear wave equation of the type ¶2 ++g¶ - D ()= () Î W > t u t uugu fx, x , t 0 , (8.5) u ==0, u u ()x , ¶ u =u ()x , ¶W t = 0 0 t t = 0 1 serves as the main example. Here g > 0 , fx()Î C¥ ()W and the conditions set on the function gs() from C¥ ()R are such that we can take gu()= sinu or gu()= = u2 p +1 , where p = 012,,,¼ for d = dim W £ 2 and p = 01, for d = 3 .

In this example the classes LmR, are nonempty for all m . Other examples will be given in Chapter 4. = We fix an integer NPP and assume N to be the projector in H onto the subspace generated by the first N eigenvectors of the operator A . Let QIP= - . If we apply the projectors PQ and to equation (8.1), then we obtain the following system of two equations for pt()= PU() t and qt()= Qu() t : ¶2 ++g¶ = ()+ t p t pAp PB p q , (8.6) ¶2 ++g¶ = ()+ t q t qAq QB p q . The reasoning below is formal. Its goal is to obtain an iteration scheme for the determination of an approximate IM. We assume that system (8.6) has an invariant manifold of the form \ \ \ \ M = {}()phpp+ (), ; p + lpp(), : pp, Î PH (8.7) 202 Inertial Manifolds

C ()12 ´ ´ h in the phase space DA H . Here hl and are smooth mappings from PHPH a into QD() A . If we substitute qt()=¶hpt()(), ¶pt() and qt()= lpt()(), ¶pt() p t t t t in the second equality of (8.6), then we obtain the following equation: e r áñdd ; \ ++áñ; -g \ - + ()+ (), \ plp p\ l p Ap PB p h p p \ \ \ 3 + g lpp(), + Ah() p, p = QB() p+ h() p, p . The compatibility condition ()¶(), ¶ () = ()(), ¶ () lpt t pt t hpt t pt gives us that \ \ \ \ ()d, = áñd; + áñ\ ; -g - + ()+ (), lpp p hp p h p Ap PB p h p p . d d \ (), \ Hereinafter p f and p ffpp are the Frechét derivatives of the function with \ áñdd ; áñ; respect to pp and ; p fw and p\ fw are values of the corresponding derivatives on an element w . Using these formal equations, we can suggest the following iteration process to {}; determine the class of functions hk lk giving the sequence of approximate IMs with the help of (8.7): (), \ = ()g+ (), \ - ()d, \ - áñ; \ - Ahk pp QB p hk - 1 pp ln ()k pp p lk - 1 p - áñd ; -g \ - + ()+ (), \ p\ lk - 1 p Ap PB p hk - 1 pp , (8.8) where k =n123,,,¼ and the integers ()k should be choosen such that k -£1 ££n () (), \ k k . Here lk pp is defined by the formula ()d, \ = áñd; \ + áñ; -g \ - + ()+ (), \ lk pp p hk - 1 p p\ hk - 1 p Ap PB p hk - 1 pp ,(8.9) where k = 123,,,¼ . We also assume that (), \ ºº(), \ h0 pp l0 pp 0 .(8.10)

(), \ ()n, \ ()= Exercise 8.1 Find the form of h1 pp and l1 pp for 1 0 and for n ()1 = 1 .

The following assertion contains information on the smoothness properties of the

functions hn and ln which will be necessary further.

Theorem 8.1. {}; Assume that the class of functions hn ln is defined according to (8.8)–(8.10). Then for each nh the functions n and ln belong to the class Cm as mappings from PH´ PH into QH and for all integers ab, ³ 0 such that ab+ £ m the estimates Approximate Inertial Manifolds for Second Order in Time Equations 203

áñab, (), \ ; ,,¼ ; \ ,,¼ \ £ AD hn pp w1 wa w1 wb a b £ × 12 \ Cab,,R Õ Awi Õ A wi , (8.11) i = 1 i = 1

12 áñab, (), \ ; ,,¼ ; \ ,,¼ \ £ A D ln pp w1 wa w1 wb a b £ × 12 \ Cab,,R Õ Awi Õ A wi (8.12) i = 1 i = 1 \ \ are valid for all pp and from PH such that Ap£ R and A12 p £ R . Hereinafter Dab, ff is the mixed Frechét derivative of the function of the \ order a with respect to p and of the order b with respect to p ; the values \ a =b= wj and wj are from PH . Moreover, if 0 or 0 , then the corresponding products in (8.11) and (8.12) should be omitted.

Proof. We use induction with respect to n . It follows from (8.10) and (8.2) that estimates (8.11) and (8.12) are valid for n = 01, . Assume that (8.11) and (8.12) hold for all nk£ - 1 . Then the following lemma holds.

Lemma 8.1. \ \ Let Fn ()pp, = Bp()+ hn ()pp, and let ab, ()= áñab, (), \ ; ,,¼ ; \ ,,¼ \ Fn w D Fn pp w1 wa w1 wb . Then for n £abk - 1 and for all integers ,ab³ 0 such that + £ m the estimate a b \ ab, ()£ × 12 Fn w CAwÕ i Õ A wi (8.13) i = 1 i = 1 \ \ Î £ 12 \ £ holds, where wj ,wj ,pp ,PH andAp R ,A p R .

Proof. ab, It is evident that Fn ()w is the sum of terms of the type s ()= áñ()s ()+ (), \ ; ,,¼ ³ Bn y B phn pp y1 ys ,.s 0

Here ys is one of the values of the form: = + áñd ; y* ws p hn ws , = áñst, ; ,,¼ ; \ ,,¼ \ y** D hn w1 wa w1 wb . 204 Inertial Manifolds

C h Equation (8.2) implies that a s p s t () £ 12 Bn y CR Õ A yj . e r j = 1 3 Therefore, the induction hypothesis gives us (8.13).

Let us prove (8.12). The induction hypothesis implies that it is sufficient to estimate the derivatives of the second term in the right-hand side of (8.9). It has the form \ áñd \ ; (), p hk - 1 Dk pp ,(8.14) where (), \ = - g\ - + ()+ (), \ Dk pp p Ap PB p hk - 1 pp . The Frechét derivatives of value (8.14) ab, \ \ \ áñáñd \ ; (), ; ,,¼ ; ,,¼ D p hk - 1 Dk pp w1 wa w1 wb are sums of the terms of the type ()st, º áñst, +1 (), \ ; ,,¼ ; \ ,,¼ \ , G D h - pp w w wi wi yst, , k 1 j1 js 1 t where as-bt, - \ \ \ yst, = áñD D ()pp, ; ww ,,¼ ww ; wr ,,¼ wr . k 1 as- 1 bt- Here 0 ££sa ,0 ££tb and the sets of indices possess the following properties: {}w,,¼ ÇÆ{},,¼w = j1 js 1 as- , {}w,,¼ È {},,¼w = {},,¼a , j1 js 1 as- 12 ; {}r,,¼ ÇÆ{},,¼r = i1 it 1 bt- , {}r,,¼ È {},,¼r = {},,¼b , i1 it 1 bt- 12 . The induction hypothesis implies that s t \ AG()st, £ CAw× A12 w × A12 y . Õ jq Õ iq st, q = 1 q = 1 Using the induction hypothesis again as well as Lemma 8.1 and the inequality 12 £ l12 A Ph N Ph , we obtain an estimate of the following form (if sa=tb or = , then the corresponding product should be considered to be equal to 1): as- bt- 12 12 12 \ A y £ C ()1 + l Aw × A wr . st, N Õ wq Õ q q = 1 q = 1 Approximate Inertial Manifolds for Second Order in Time Equations 205

l Hereinafter k is the k -th eigenvalue of the operator A . Thus, it is possible to state that ()st, £ ()+ l12 × 12 \ AG C 1 N Õ Awi Õ A wi .(8.15) i i Using the inequality £ l-s s > Qh N +1 A Qh ,,s 0 (8.16) and equation (8.15) it is easy to find that estimates (8.12) are valid for nk= . If we use (8.8), (8.12) and follow a similar line of reasoning, we can easily obtain (8.11). Theorem 8.1 is proved.

Theorem 8.1 and equation (8.4) imply the following lemma.

Lemma 8.2. () ³ Assume that ut is a solution to problem (8.1) lying in LmR, ,m 1 . Let pt()= Pu() t and let ()= ()(), ¶ () ()= ()(), ¶ () qs t hs pt t pt ,.qs t ls pt t pt (8.17) Then the estimates 12 ()j ()2 + ()j ()2 £ A qs t Aqs t CRm, with 0 ££jm-1 and ()m ()2 + 12 ()m ()2 £ qs t A qs t CRm, are valid for t large enough.

Proof. ()j () It should be noted that qs t is the sum of terms of the form ()i ()i ()t + 1 ()t +1 áñab, (), ¶ ,,,1 () ¼ a (); 1 (),, ¼ b () D hs p tp p t p t p t p t , ab ,,¼ t,,¼t where , ,i1 ia , 1 b are nonnegative integers such that ££ab+ +++++¼ t ¼t= 1 j ,.i1 ia 1 b j ()i () Similar equation also holds for qs t . Further one should use Theorem 8.1 and the estimates ()k +1 ()2 ++12 ()k ()2 ()n -1 ()2 £ 2 ³ ££ p t A p t Ap t R ,,tt* 1 km , which follow from (8.4).

Let us define the induced trajectories of the system by the formula ()= ()(); () Us t us t us t , where s = 012,,,¼ and ()= ()+ () ()= ¶ ()+ () us t pt qs t ,.(8.18)us t t pt qs t 206 Inertial Manifolds

C ()= () () () () h Here pt Pu t ,ut is a solution to problem (8.1); qs t and qs t are defined a with the help of (8.17). Assume that ut() lies in L , . Then Lemma 8.2 implies p mR t that the induced trajectories can be estimated as follows: e () () r 12 j () + j () £ ££ - A us t Aus t CRs, ,;0 jm1 3 ()m ()2 + 12 ()m ()2 £ us t A us t CRs, for t large enough. Using (8.3), (8.4), and the last estimates, it is easy to prove the following assertion (do it yourself).

Lemma 8.3. Let ()= ()()+ () - ()()+ () Es t Bpt qt Bpt qs t . Then j ()j () £ 12 ()()i ()- ()i () Es t CRj, å A q t qs t i = 0 for j = 01,,¼ ,m and for t large enough.

The main result of this section is the following assertion.

Theorem 8.2. () ³ Let ut be a solution to problem (8.1) lying in LmR, with m 2 . (), \ (), \ Assume that hn pp and ln pp are defined by (8.8)–(8.10). Then the estimates ¶ j ()()- () £ l-n2 A t ut un t CnR, N +1 , (8.19)

12 ¶ j ()¶ ()- () £ l-n2 A t tut un t CnR, N +1 , (8.20) are valid for nm£ -1 and for t large enough. Here 0 ££jmn- -1 , () () l ()+ un t and un t are defined by (8.18), and N +1 is the N 1 -th eigenvalue of the operator A .

Proof. Let us consider the difference between the solution ut() and the trajectory induced by this solution: c ()=c()- () ()= ¶ ()- () ³ s t ut us t ,,,s t tut us t s 0 () () c()= () where us t and us t are defined by formula (8.18). Since 0 t qt , equation (8.4) implies that () 12 c()j +1 () + c j () £ = ,,,¼ , - A 0 t A 0 t C ,,(8.21)j 012 m 1 Approximate Inertial Manifolds for Second Order in Time Equations 207 for t large enough. Equations (8.8)–(8.10) also give us that c ()= - c²()- gc¢()+ () A 1 t 0 t 0 t QE0 t . We use Lemma 8.3 and equation (8.21) to find that c()j () £ l-12 = ,,¼ , - A 1 t C N +1 ,,j 01 m 2 for tn large enough. Therefore, equation (8.19) holds for = 01, and for t large enough. From equations (8.6), (8.8), and (8.9) it is easy to find that c = -g¶¶ c - c - gdáñ; - A k t k -1 t n()k -1 p\ hn ()k -1 PEn ()k -1 - áñd ; + p\ lk -1 PEk -1 QEk -1 and c = ¶ c + áñd ; k t k -1 p\ hk -1 PEk -1 .(8.22)

Lemma 8.4. The estimates j ¶ j áñd ; £ l12 12 c()s () A t p\ hn PEn C N å A n t (8.23) s = 0 and j 12 ¶ j áñd ; £ l12 12 c()s () A t p\ ln PEn C N å A n t (8.24) s = 0 are valid for t large enough and for each n ³ 0 , where j = 01,,¼ ,m -1 .

Proof. = =¶j áñd ; Let fn hn or fn ln . It is clear that the value t p\ fn PEn is the algebraic sum of terms of the form: g g s s áñab, +1 ; 1,,¼ a ; 1,,¼ b ,¶s D fn p p p p t PEn . Therefore, Theorem 8.1 and Lemma 8.3 imply (8.23) and (8.24). Lemma 8.4 is proved.

We use Lemmata 8.3 and 8.4 as well as inequality (8.16) to obtain that j + 2 j +1 ìü c()j () £ l-1 c()s () + c()s () + A k t ckj, N +1 íýå A k -1 t å A n ()k -1 t îþs = 0 s = 0 j + l-12 c()s () dkj N +1 å A k -1 t , (8.25) s = 0 = ,,¼ , - where j 01 m 2 and the numbers ckj, and dkj, do not depend on N . 208 Inertial Manifolds

C £ - h If we now assume that (8.19) holds for nk1 , then equation (8.25) implies a (8.19) for nk= and for km£ - 1 . Using (8.22) and (8.23) we obtain equation p t (8.20). Theorem 8.2 is proved. e r Corollary 8.1 3 (), \ = (), \ Let the manifold Mn have the form (8.7) with hpp hn pp and (), \ = (), \ ()= ()(); \ () () lpp ln pp. We also assume that Ut ut u t , where ut is the solution to problem (0.1) from the class LmR, . Then -n¤ 2 dist ()Ut(), M £ C l ,.n = 012,,¼m - 1 DA´ DA12¤ n n N +1

Thus, the thickness of the layer that attracts the trajectories in the phase space has l the power order with respect to N +1 unlike the semilinear parabolic equations of Section 6.

Example 8.1 Let us consider the nonlinear wave equation (8.5). Let d = dimW £ 2 . We assume the following (cf. [18]) about the function gs() : s lim s-1 òg()ss d ³ 0 ; s ® ¥ 0 > there exists C1 0 such that æös -1 ç÷()- ()ss ³ lim s ç÷sg s C1 g d 0 ; s ® ¥ ò èø0 for any m there exists b ()m > 0 such that ()m () £ ()+ b ()m g s C2 1 s .(8.26) () > Under these assumptions the solution ut lies in LmR, for R 0 large enough if and only if the initial data satisfy some compatibility conditions [18]. Moreover, the global attractor ) of system (8.5) exists and any trajectory lying in ) possesses properties (8.4) for all t Î R and k = 12,,¼ , [18]. It is easy to see that Theorem 8.2 is applicable here (the form of AB ,(). and H is evident in this case). In particular, Theorem 8.2 gives us that for a trajectory ()= ()(); ¶ ()\ ) Ut ut t ut of problem (8.5) which lies in the global attractor the estimate ìü12 j 2 12 j 2 -n2 íýA¶ ()ut()- u ()t + A ¶ ()¶ ut()- u ()t £ C ,, l + îþt n t t n nRj N 1 = ,,¼ = ,,¼ Î () holds for all n 12 , all j 12 , and all t R . Here un t and () un t are defined with the help of (8.18). Therewith Idea of Nonlinear Galerkin Method 209

{}(), Î) £ l-n2 = ,,¼ sup dist U Mn : U cn N +1 ,,(8.27)n 12 = (), \ = (), \ where Mn is a manifold of the type (8.7) with hhn pp and lln pp . (), Here dist U Mn is the distance between U and Mn in the space DA()´ DA()12 . Equation (8.27) gives us some information on the location of the global attractor in the phase space.

Other examples of usage of the construction given here can be found in papers [17] and [19] (see also Section 9 of Chapter 4).

§ 9 Idea of Nonlinear Galerkin Method

Approximate inertial manifolds have proved to be applicable to the computational study of the asymptotic behaviour of infinite-dimensional dissipative dynamical systems (for example, see the discussion and the references in [8]). Their usage leads to the appearance of the so-called nonlinear Galerkin method [20] based on the re- placement of the original problem by its approximate inertial form. In this section we discuss the main features of this method using the following example of a second order in time equation of type (8.1): 2u u u d ++g d Au = Bu(),,u = u d = u .(9.1) 2 dt t = 0 0 t 1 dt d t = 0 If all conditions on A and B(). given in the previous section are fulfilled, then {}; Theorem 8.2 is valid. It guarantees the existence of a family of mappings hk lk from PH´ PH into QH possessing the properties: º (), r º (), r = , 1) there exist constants Mj Mj n and Lj Lj n , j 12 , such that (), \ £ 12 (), \ £ Ahn p0 p0 M1 ,e,(9.2)A ln p0 p0 M2

()(), \ - (), \ £ æö()- + 12 ()\ - \ Ahn p1 p1 hn p2 p2 L1èøAp1 p2 A p1 p2 ,(9.3)

12 ()(), \ - (), \ £ æö()- + 12 ()\ - \ A ln p1 p1 ln p2 p2 L1èøAp1 p2 A p1 p2 (9.4) \ for all pj and pj from PH such that 2 +r12 \ 2 £ 2 =r, > Apj A pj ,,;j 01 0 () ³ 2) for any solution ut to problem (9.1) which lies in LmR, for m 2 the estimate 210 Inertial Manifolds

C h 12 ìü2 -n2 a ()()- () 2 + 12 ()¶ ()- () £ l íýAut un t A t ut un t CnR, N +1 (9.5) p îþ t e is valid (see Theorem 8.2) for all nm£ - 1 and t large enough. Here r u ()t = pt()+ h ()pt(), ¶pt() , 3 n n t (9.6) ()= ¶ ()+ ()(), ¶ () un t t pt ln pt t pt , l ()+ N +1 is the N 1 -th eigenvalue of AR , and is the constant from (8.4). {}; The family hk lk is defined with the help of a quite simple procedure (see (8.8) and (8.9)) which can be reduced to the process of solving of stationary equations of the type A vg= in the subspace QH . Moreover, (), \ ºº(), \ (), \ = -1 () (), \ º h0 pp l0 pp 0 ,,.(9.7)h1 pp A QB p l1 pp 0 In particular, estimates (9.5) and (9.6) mean (see Corollary 8.1) that trajectories ()= ()(); ¶ () Ut ut t ut of system (9.1) are attracted by a small (for N large enough) vicinity of the manifold = {}()+ (), \ ; \ + (), \ , \ Î Mn phn pp p ln pp : pp PH .(9.8) {}(), \ The sequence of mappings hn pp generates a family of approximate inertial forms of problem (9.1): ¶2 ++g¶ = ()+ (), ¶ t p t pAp PB p hn p t p .(9.9) A finite-dimensional dynamical system in PH which approximates (in some sense) the original system corresponds to each form. For n = 0 equation (9.9) transforms into the standard Galerkin approximation of problem (9.1) (due to (9.7)). If n > 0 , then we obtain a class of numerical methods which can be naturally called the nonlinear Galerkin methods. However, we cannot use equation (9.9) in the computa- (), \ tional study directly. The point is that, first, in the calculation of hn pp we have to solve a linear equation in the infinite-dimensional space QH and, second, we can lose the dissipativity property. Therefore, we need additional regularization. It can (), \ (), \ be done as follows. Assume that fn pp stands for one of the functions hn pp (), \ or ln pp . We define the value 12 *(), \ ºc(), \ = æö-1æö2 + 12 \ 2 (), \ fn pp fNMn,, pp èøR èøAp A p PM fn pp,(9.10)

where c ()s is an infinitely differentiable function on R+ such that a)0 ££c()s 1 ; b) c()s = 1 for 0 ££s 1 ; c) c()s = 0 for s ³ 2 ; R is the radius of dissipativity = (see (8.4) for k 0 ) of system (9.1); PM is the orthoprojector in H onto the subspace generated by the first MAMN eigenvectors of the operator , > . We consider the following NP -dimensional evolutionary equation in the subspace N H : Idea of Nonlinear Galerkin Method 211

¶2 * ++g¶ * * = ()* + * ()*, ¶ * t p t p Ap PN Bp hn p t p , (9.11) p* ==P u , ¶ p* P u . t = 0 N 0 t t = 0 N 1

Exercise 9.1 Prove that problem (9.11) has a unique solution for t > 0 and ´ the corresponding dynamical system is dissipative in PN HPN H .

We call problem (9.11) a nonlinear Galerkin ()nNM,, -approximation of problem (9.1). The following assertion is valid.

Theorem 9.1. (), \ (), \ Assume that the mappings hn pp and ln pp satisfy equations (9.2)–(9.5) for nm£ -1 and for some m ³ 2 . Moreover, we assume that > * * (9.5) is valid for all t 0 . Let hn and ln be defined by (9.10) with the help of hn and ln and let * ()= *()+ * ()*(), ¶ *() un t p t hn p t t p t , * ()= ¶ *()+ * ()*(), ¶ *() un t t p t ln p t t p t , where p*()t is a solution to problem (9.11). Then the estimate ìü12 £ 12 ()()- * () 2 + ¶ ()- * ()2 íýA ut un t t ut un t îþ £ ()ba l-()n +1 ¤ 2 + a l-12¤ () 1 N +1 2 M +1 exp t (9.12) () holds, where ut is a solution to problem (9.1) which lies in LmR, for m ³ 2 and possesses property (8.4) for k = 1 and for all t > 0 . Here n £ £a- a b m 1 , 1 , 2 and are positive constants independent of MN and , l k is the k -th eigenvalue of the operator A .

Proof. ()= () Let pt PN ut . We consider the values ()- * ()= ()- *()++[]()- ()(), ¶ () ut un t pt p t QN ut hn pt t pt + []()(), ¶ () - * ()*(), ¶ *() hn pt t pt hn p t t p t and ¶ ()-¶* ()= ()()- *() ++[]¶ ()- ()(), ¶ () t ut un t t pt p t QN t ut ln pt t pt + []()(), ¶ () - * ()*(), ¶ *() ln pt t pt ln p t t p t . The equalities ()(), ¶ () = * ()(), ¶ () PM hn pt t pt hn pt t pt 212 Inertial Manifolds

C h and a ()(), ¶ () = * ()(), ¶ () p PM ln pt t pt ln pt t pt t e are valid for the class of solutions under consideration. Therefore, we use (9.5) to r find that 3 12 ()()- * () £ A ut un t C C £ æö12 ()¶()- *() + ()- ¶ *() ++2 nR, C1 èøA pt p t t pt t p t (9.13) l12 l()n +1 ¤ 2 M +1 N +1 and ¶ ()- ¶ * () £ t ut t un t C C £ æö12 ()¶()- *() + ()- ¶ *() ++4 nR, C3 èøA pt p t t pt t p t . (9.14) l12 l()n +1 ¤ 2 M +1 n +1 Therefore, we must compare the solution p*()t to problem (9.11) with the value ()= () pt PN ut which satisfies the equation ¶2 ++g¶ = ()+ t p t pAp QN Bp QN u (9.15) with the same initial conditions as the function p*()t . Let rt()= pt()- p*()t . Then it follows from (9.11) and (9.15) that ¶2 ()++ g¶ () ()= (),,* t rt t rt Ar t Ft p u , (9.16) ()==, ¶ () r 0 0 t r 0 0 , where (),,* = []()() - ()* () Ft p u QN But Bun t . Due to the dissipativity of problems (9.11) and (9.15) we use (9.13) to obtain

12 (),,* £ æö12 ()2 + \ ()2 ++l-()n +1 ¤ 2 l-12 Ft p u CR èøA rt r t CnR, N +1 C M +1 for the class of solutions under consideration. Therefore, equation (9.16) implies that

1 d æö\ 2 2 æö\ 2 2 -()n +1 -1 r ()t + A12rt() £ C r ()t + A12rt() ++C , l C l . 2 dt èøRèønR N +1 M +1 Hence, Gronwall’s lemma gives us that

\ ()2 + 12 ()2 £ ()l-()n +1 + l-1 CR t r t A rt CnR, N +1 C M +1 e . This and equations (9.13) and (9.14) imply estimate (9.12). Theorem 9.1 is proved.

If we take n = 0 and NM= in Theorem 9.1, then estimate (9.12) changes into the accuracy estimate of the standard Galerkin method of the order N . Therefore, if the Idea of Nonlinear Galerkin Method 213

l £ ln + 1 parameters NM , , and n are compatible such that M +1 N +1 , then the error of the corresponding nonlinear Galerkin method has the same order of smallness as in the standard Galerkin method which uses M basis functions. However, if we use the nonlinear method, we have to solve a number of linear algebraic systems of the order MN- and the Cauchy problem for system (9.11) which consists of N equations. (), \ In particular, in order to determine the value h1 pp we must solve the equation (), \ = ()- () A h1 pp PM PN QB p = l £ l2 for n 1 and choose the numbers NM and such that M +1 N +1 . Moreover, l @ss ()+ () > ® ¥ ifk c0 k 1 o 1 , 0 , as k , then the values NM and must be com- 2 patible such that Mc£ s N . We note that Theorem 9.1 as well as the corresponding variant of the nonlinear Galerkin method can be used in the study of the asymptotic properties of solutions to the nonlinear wave equation (8.5) under some conditions on the nonlinear term gu(). Other applications of Theorem 9.1 can also be pointed out. 214 Inertial Manifolds

C h References a p t e 1. FOIAS C., SELL G. R., TEMAM R. Inertial manifolds for nonlinear evolutionary r equations // J. Diff. Eq. –1988. –Vol. 73. –P. 309–353. 2. CHOW S.-N., LU K. Invariant manifolds for flows in Banach spaces // J. Diff. Eq. 3 –1988. –Vol. 74. –P. 285–317. 3. MALLET-PARET J., SELL G. R. Inertial manifolds for reaction-diffusion equations in higher space dimensions. // J. Amer. Math. Soc. –1988. –Vol. 1. –P. 805–866. 4. CONSTANTIN P., FOIAS C., NICOLAENKO B., TEMAM R. Integral manifolds and inertial manifolds for dissipative partial differential equations. –New York: Springer, 1989. 5. TEMAM R. Infinite-dimensional dynamical systems in Mechanics and Physics. –New York: Springer, 1988. 6. MIKLAV«I« M. A sharp condition for existence of an inertial manifold // J. of Dyn. and Diff. Eqs. –1991. –Vol. 3. –P. 437–456. 7. ROMANOV A. V. Exact estimates of dimention of inertial manifold for nonlinear parabolic equations // Izvestiya RAN. –1993. –Vol. 57. –# 4. –P. 36–54 (in Russian). 8. CHUESHOV I. D. Global attractors for non-linear problems of mathematical physics // Russian Math. Surveys. –1993. –Vol. 48. –# 3. –P. 133–161. 9. CHUESHOV I. D. Introduction to the theory of inertial manifolds. –Kharkov: Kharkov Univ. Press., 1992 (in Russian). 10. CHEPYZHOV V. V., GORITSKY A. YU. Global integral manifolds with exponential tra- cking for nonautonomous equations // Russian J. Math. Phys. –1997. –Vol. 5. –#l. 11. MITROPOLSKIJ YU.A., LYKOVA O. B. Integral manifolds in non-linear mechanics. –Moskow: Nauka, 1973 (in Russian). 12. HENRY D. Geometric theory of semilinear parabolic equations. Lect. Notes in Math. 840. –Berlin: Springer, 1981. 13. BOUTET DE MONVEL L., CHUESHOV I. D., REZOUNENKO A. V. Inertial manifolds for retarded semilinear parabolic equations // Nonlinear Analysis. –1998. –Vol. 34. –P. 907–925. 14. BARIS YA.S., LYKOVA O. B. Approximate integral manifolds. –Kiev: Naukova dumka, 1993 (in Russian). 15. CHUESHOV I. D. Approximate inertial manifolds of exponential order for semilinear parabolic equations subjected to additive white noise // J. of Dyn. and Diff. Eqs. –1995. –Vol. 7. –# 4. –P. 549–566. 16. MORA X. Finite-dimensional attracting invariant manifolds for damped semilinear wave equations // Res. Notes in Math. –1987. –Vol. 155. –P. 172–183. 17. CHUESHOV I. D. On a construction of approximate inertial manifolds for second order in time evolution equations // Nonlinear Analysis, TMA. –1996. –Vol. 26. –# 5. –P. 1007–1021. 18. GHIDAGLIA J.-M., TEMAM R. Regularity of the solutions of second order evolution equations and their attractors // Ann. della Scuola Norm. Sup. Pisa. 1987. –Vol. 14. –P. 485–511. 19. CHUESHOV I. D. Regularity of solutions and approximate inertial manifolds for von Karman evolution equations // Math. Meth. in Appl. Sci. –1994. –Vol. 17. –P. 667–680. 20. MARION M., TEMAM R. Nonlinear Galerkin methods // SIAM J. Num. Anal. –1989. –Vol. 26. –P. 1139–1157. Chapter 4

The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

Contents

. . . . References ...... 281

In this chapter we use the ideas and the results of Chapters 1 and 3 to study in details the asymptotic behaviour of a class of problems arising in the nonlinear theory of oscillations of distributed parameter systems. The main object is the following second order in time equation in a separable Hilbert space H : 2 d +++g d 2 æö12¤ 2 += () u uAuMAèøu Au Lu pt , (0.1) dt2 dt u u = u , d = u , (0.2) t = 0 0 t 1 d t = 0 where AHMs is a positive operator with discrete spectrum in , () is a real function (its properties are described below), LHpt is a linear operator in , () is a given bounded function with the values in H , and g is a nonnegative parameter. The problem of type (0.1) and (0.2) arises in the study of nonlinear oscillations of a plate in the supersonic flow of gas. For example, in Berger’s approach (see [1, 2]), the dynamics of a plate can be described by the following quasilinear partial differential equation: æö ¶2 +++g¶ D2 ç÷G - Ñ 2 D +r¶ = (), t u t u u u dx u x u px t , (0.3) èøò 1 W ÎW(), ÌÌ2 > xx1 x2 R , t 0 with boundary and initial conditions of the form ==D = () ¶ = () u ¶W u ¶W 0 , u = u0 x , tu u1 x . (0.4) t 0 t = 0 Here DWg is the Laplace operator in the domain ; >r0 , ³G0 , and are con- (), () () stants; and px t , u0 x , and u1 x are given functions. Equations (0.3)–(0.4) describe nonlinear oscillations of a plate occupying the domain W on a plane which is located in a supersonic gas flow moving along the x1 -axis. The aerodynamic pressure on the plate is taken into account according to Ilyushin’s “piston” theory (see, r¶ r e. g., [3]) and is described by the term x1u . The parameter is determined by the velocity of the flow. The function ux(), t measures the plate deflection at the point xt and the moment . The boundary conditions imply that the edges of the plate are hinged. The function px(), t describes the transverse load on the plate. The parameter G is proportional to the value of compressive force acting in the plane of the plate. The value g takes into account the environment resistance. Our choice of problem (0.1) and (0.2) as the base example is conditioned by the following circumstances. First, using this model we can avoid significant technical difficulties to demonstrate the main steps of reasoning required to construct a solution and to prove the existence of a global attractor for a nonlinear evolutionary second order in time partial differential equation. Second, a study of the limit regimes of system (0.3)–(0.4) is of practical interest. The point is that the most important (from the point of view of applications) type of instability which can be found in the system under consideration is the flutter, i.e. autooscillations of a plate subjected to 218 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h aerodynamical loads. The modern look on the flutter instability of a plate is the fol- a lowing: there arises the Andronov-Hopf bifurcation leading to the appearance of p t a stable limit cycle in the system. However, there are experimental and numerical e data that enable us to conjecture that an increase in flow velocity may result in the r complication of the dynamics and appearance of chaotic fluctuations [4]. Therefore, 4 the study of the existence and properties of the attractor of the given problem enables us to better understand the mechanism of appearance of a nonlinear flutter.

§ 1 Spaces

As above (see Chapter 2), we use the scale of spaces s generated by a positive operator AH with discrete spectrum acting in a separable Hilbert space . We remind (see Section 2.1) that the space s is defined by the equation ìü¥ ¥ º ()s = 2 l2 s < ¥ s DA íýv : å ck ek : å ck k , îþ k = 1 k = 1 {} where ek is the orthonormal basis of the eigenelements of the operator AH in , l ££l ¼ 1 2 are the corresponding eigenvalues and s is a real parameter = = < (fors 0 we have s H and for s 0 the space s should be treated as a class of formal series). The norm in s is given by the equality ¥ ¥ 2 = l2 s = v s å ck k forvcå k ek . k = 1 k = 1 2 (), ; Further we use the notation L 0 T s for the set of measurable functions [], on the segment 0 T with the values in the space s such that the norm æöT 12¤ ç÷ v = vt()2 dt 2(), ; ç÷ò s L 0 T s èø0 is finite. The notation Lp()0, T ; X has a similar meaning for 1 ££p ¥ . We remind that a function ut with the values in a separable Hilbert space H is said to be Bochner measurable on a segment []0, T if it is a limit of a sequence of functions N ()= c () uN t å uNk, Nk, t k = 1 Î [], c () for almost all t 0 T , where uNk, are elements of H and Nk, t are the characteristic functions of the pairwise disjoint Lebesgue measurable sets ANk, . One Spaces 219 can prove (see, e.g., the book by K. Yosida [5]) that for separable Hilbert spaces under consideration a function ut() is measurable if and only if the scalar function ()(), Î () ut h H is measurable for every hH . Furthermore, a function ut is said to be Bochner integrable over []0, T if T ut()- u ()t 2 t ® 0 ,,N ® ¥ ò N H d 0 {}() where uN t is a sequence of simple functions defined above. The integral of the function ut over a measurable set S Ì []0, T is defined by the equation T () = c ()t ()t t ut dt lim S uN d , ò N ® ¥ ò S 0 c ()t where S is the characteristic function of the set S and the integral of a simple function in the right-hand side of the equality is defined in an obvious way. For the function with the values in Hilbert spaces most facts of the ordinary Le- besgue integration theory remain true.

Exercise 1.1 Let ut() be a function on []0, T with the values in a separable Hilbert space H . If there exists a sequence of measurable func- () ()® () () tions un t such that un t ut almost everywhere, then ut is also measurable.

Exercise 1.2 Show that a measurable function ut() with the values in H is integrable if and only if ut() Î L10, T . Therewith

òut()dt £ ò ut() dt B B for any measurable set B Ì []0, T .

Exercise 1.3 Let a function ut() be integrable over []0, T and let B be a measurable set from []0, T . Show that æö ()()t , t = ç÷()t t, ò u h H d ç÷òu d h èø B B H

for any hHÎ . 2 (), ; Exercise 1.4 Show that the space L 0 T s can be described as a set of series ¥ = () ht å ck t ek , k = 1 220 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C () h where ck t are scalar functions that are square-integrable over a []0, T and such that p t ¥ T e l2 s []()2 < ¥ r å k ò ck t dt . (1.1) 4 k = 1 0 (), ; [], Below we also use the space C 0 T s of strongly continuous functions on 0 T with the values in s and the norm v (), ; = max vt() . C 0 T s t Î []0, T s

() Exercise 1.5 Let ut be a function with the values in s integrable over []0, T . Show that the function t vt()= ò u()t dt 0 (), ; () lies in C 0 T s . Moreover, vt is an absolutely continuous e >d> function with the values in s , i.e. for any 0 there exists 0 []a , b Ì [], such that for any collection of disjoint segments k k 0 T the condition å ()b - a < d implies that k k k v()b - v()a < e . å k k s k

Exercise 1.6 Show that for any absolutely continuous function vt() on [], () 0 T with the values in s there exists a function ut with the [], values in s such that it is integrable over 0 T and t vt()= v()0 + ò u()t dt ,.t Î []0, T 0 (Hint: use the one-dimensional variant of this assertion).

The space ìü\ W = íývt(): vt()Î L2()0, T ; , v()t Î L2 ()0, T ; H (1.2) T îþ1 with the norm æö\ 12¤ v = v 2 + v 2 W èø2 (), ; L2()0, T ; H T L 0 T 1 \ plays an important role below. Hereinafter the derivative v()t = dv¤ dt stands for a function integrable over []0, T and such that Spaces 221

t \ vt()= hv+ ò ()t dt 0 almost everywhere for some hHÎ (see Exercises 1.5 and 1.6). Evidently, the space (), ; () WT is continuously embedded into C 0 T H , i.e. every function ut from WT lies in C ()0, T ; H and () £ max ut CuW , t Î []0, T T where C is a constant. This fact is strengthened in the series of exercises given below.

{ = Exercise 1.7 Let pm be the projector onto the span of the set ek : k =},,¼ ()Î () 1 m and let vt WT . Show that pm vt is absolutely continuous and possesses the property \ d p vt() = p v()t Î L2 ()0, T ; p . dt m m m 1

Exercise 1.8 The equations t \ p vt()2 = p vs()2 + 2 ()p v()t , p v()t t (1.3a) m 12¤ m 12¤ ò m m 12¤ d s and 2 ()ts- p vt() = m 12¤

t 2 \ = æöp v()t + 2 ()t - s ()p v()t , p v()t dt (1.3b) ò èøm 12¤ m m 12¤ s £££ ()Î are valid for any 0 stT and vt WT . Exercise 1.9 Use (1.3) to prove that sup p vt() £ C v (1.4a) m 12¤ T W t Î []0, T T and sup ()p - p vt() £ C ()p - p v . (1.4b) m k 12¤ T m k W t Î []0, T T

Exercise 1.10 Use (1.4) to prove that WT is continuously embedded into (), ; C 0 T 12¤ and () £ max vt 12¤ CT v W . t Î []0, T T 222 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h The following three exercises result in a particular case of Dubinskii’s theorem a (see Exercise 1.13). p t ¥ e Exercise 1.11 Let {}h ()t be an orthonormal basis in L2()0, T ; R con- r k k = 0 sisting of the trigonometric functions 4 p p ()= 1 ()= 2 2 k ()= 2 2k h0 t ,,,h2 k -1 t sin x h2 k t cos x T T T T T = ,,¼ ()Î 2 (), ; k 12 . Show that ft L 0 T s if and only if ¥ ¥ ()= () ft å å ckj hk t ej (1.5) k = 0 j = 1 and ¥ ¥ l2 s 2 < ¥ å å j ckj . k = 0 j = 1

Exercise 1.12 Show that the space WT can be described as a set of series of the form (1.5) such that ¥ æö2 + l2 2 < ¥ å èøk j ckj . kj, = 1

Exercise 1.13 Use the method of the proof of Theorem 2.1.1 to show that 2(), ; WT is compactly embedded into the space L 0 T s for any s < 1 . (), ; Exercise 1.14 Show that WT is compactly embedded into C 0 T H . Hint: use Exercise 1.10 and the reasoning which is usually applied to prove the Arzelà theorem on the compactness of a collection of scalar continuous functions.

§ 2 Auxiliary Linear Problem

In this section we study the properties of a solution to the following linear problem:

ì 2 u u ï d +++g d A2ubt()Au = ht() , (2.1) ï dt2 dt í ï du u = u , = u . (2.2) ï t = 0 0 dt 1 î t = 0 Auxiliary Linear Problem 223

() Here A is a positive operator with discrete spectrum. The vectors ht , u0 , u1 as well as the scalar function bt() are given (for the corresponding hypotheses see the assertion of Theorem 2.1). The main results of this section are the proof of the theorem on the existence and uniqueness of weak solutions to problem (2.1) and (2.2) and the construction of the evolutionary operator for the system when ht()º 0 . In fact, the approach we use here is well-known (see, e.g., [6] and [7]). A weak solution to problem (2.1) and (2.2) on a segment []0, T is a func- ()Î ()= tion ut WT such that u 0 u0 and the equation T T \ \ - ò ()u()t + gut(), v ()t dt +ò ()Au() t + bt()ut(), Av() t dt = 0 0 T = ()+ g , ()+ ()(), () u1 u0 v 0 ò ht vt dt (2.3) 0 ()Î ()= \ holds for any function vt WT such that vT 0 . As above, u stands for the derivative of ut with respect to .

Exercise 2.1 Prove that if a weak solution ut() exists, then it satisfies the equation ()\ ()+ g (), = ()+ g , - u t ut w u1 u0 w

t t - ò()tAu()t + b()t u()t , Aw d + ò()th()t , w d (2.4) 0 0 Î ()= T jt()td × for every w 1 (Hint: take vt òt w in (2.3), where j ()t is a scalar function from C []0, T ).

Theorem 2.1 Î Îg ³ () Let u0 1 , u1 0 , and 0 . We alalsssoo assume that bt is a bounded [], ()Î ¥ (), ; continuous function on 0 T and ht L 0 T 0 , where T is a positive number. Then problem (2.1) and (2.2) has a unique weak solution ut() on the segment []0, T . This solution possesses the properties () Î (), ; \ ()Î (), ; ut C 0 T 1 , u t C 0 T 0 (2.5) and satisfies the energy equation t t 1 æö\ \ \ u()t 2 + Au() t 2 ++g u()t 2 dt b()t ()tAu()t , u()t d = 2 èøò ò 0 0 t 1 æö\ = u 2 + Au 2 + ()th()t , u()t d . (2.6) 2 èø1 0 ò 0 224 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h Proof. a We use the compactness method to prove this theorem. At first we construct ap- p t proximate solutions to problem (2.1) and (2.2). The approximate Galerkin solution e {} r (to this problem) of the order me with respect to the basis k is considered to be the function 4 m ()= () um t å gk t ek (2.7) k = 1 satisfying the equations ()\\ ++g \ 2 +() - (), = um um A um bt Aum ht ej 0 , (2.8) ()(), = (), ()\ (), = (), = ,,¼ , um 0 ej u0 ej ,,.(2.9)um 0 ej u1 ej j 12 m ()Î 1 (), \ () Here gk t C 0 T and gk t is absolutely continuous. Due to the orthogonality {} of the basis ek equations (2.8) and (2.9) can be rewritten as a system of ordinary differential equations: \\ ++g \ l2 - ()l = ()º ()(), gk gk k gk bt k gk hk t ht ek e, ()= (), \ ()= (), = ,,¼ , gk 0 u0 ek ,,gk 0 u1 ek k 12 m .

Lemma 2.1 Assume that g > 0 , a ³ 0 , bt() is continuous, and ct() is a measurable bounded function. Then the Cauchy problem ì \\ \ ï g ++g g aa()- bt() g = ct(), t Î []0, T , í (2.10) \ ï g()0 = g , g ()0 = g î 0 1 is uniquely solvable on any segment []0, T . Its solution possesses the property æöt \ ç÷1 b t g ()t 2 + a2 gt()2 £ g2 ++a2 g2 c ()t 2 dt e 0 , t Î []0, T , (2.11) ç÷1 0 2 g ò èø0 where b = max bt() . Moreover, if bt()Î C1()0, T , ct()º 0 and for all 0 t t Î []0, T the conditions g2 \ æö - 1 a + ££bt() 1 a ,,b()t + g a - bt() ³ 0 (2.12) 2 a 2 èø4 hold, then the following estimate is valid: \ æög g ()t 2 + a2 gt()2 £ 3 ()g2 + a2 g2 exp - t . (2.13) 1 0 èø2 Auxiliary Linear Problem 225

Proof. Problem (2.10) is solvable at least locally, i.e. there exists t such that a solution exists on the half-interval [0,)t . Let us prove estimate (2.11) for the in- \ terval of existence of solution. To do that, we multiply equation (2.10) byg()t . As a result, we obtain that \ \ \ 1 d ()gg\2 + a2 g2 + g2 = ab() t gg+ ct()g . 2 dt We integrate this equality and use the equations \ \ \ \ abt()gg £ 1 max bt()æög2 + a2 g2 ,,cg £ g g2 + 1 c2 2 t èø 4g to obtain that t t \ 2 2 2 1 \ 2 2 g()t + a2 gt() £ g ++a2 g c2()t d t +b ()tg()t + a2 g()t d . 1 0 2g ò 0 ò 0 0 This and Gronwall’s lemma give us (2.11). In particular, estimate (2.11) enables us to prove that the solution gt() can be extended on a segment []0, T of arbitrary length. Indeed, let us assume the contrary. Then there exists a point t such that the solution can not be extended through it. Therewith equation (2.11) implies that \ ()2 + 2 ()2 £ (); , << < g t a gt CT g0 g1 ,.0 ttT \\ Therefore, (2.10) gives us that the derivative g()t is bounded on [)0, t . Hence, the values t t \ ()= + \\ ()t t ()= + \ ()t t g t g0 òg d , gt g0 òg d 0 0 are continuous up to the point t . If we now apply the local theorem on existence to system (2.10) with the initial conditions at the point t that are equal to \ gt() and g()t , then we obtain that the solution can be extended through t . This contradiction implies that the solution gt() exists on an arbitrary segment []0, T . Let us prove estimate (2.13). To do that, we consider the function

\ g æö\ g Vt()= 1 æög2 + aa()- bt() g2 + gg + g2 . (2.14) 2 èø2 èø2 Using the inequality \ g \ \ g - 1 g2 - g2 ££gg 1 g2 + g2 , 2g 2 2g 2 it is easy to find that the equation æö\ \ 1 g2 + a2 g2 ££Vt() 3 æög2 + a2 g2 (2.15) 4 èø 4 èø 226 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h holds under the condition a 1 1 p abt- ()³ 0 ,.a abt+g() - 2 ³ 0 t 2 2 e r Further we use (2.10) with ct()º 0 to obtain that 4 V g \ \ d = - g2 - 1 ()ab+ a g ()ab- g2 . dt 2 2 Consequently, with the help of (2.15) we get V g d + V £ 0 dt 2 under conditions (2.12). This implies that æög Vt() £ V()0 exp - t . èø2 We use (2.15) to obtain estimate (2.13). Thus, Lemma 2.1 is proved.

Exercise 2.2 Assume that g £ 0 in Lemma 2.1. Show that problem (2.10) is uniquely solvable on any segment []0, T and the estimate

()b ++2 gdt \ ()+ 2 ()2 £ æö2 + 2 2 0 + g t a gt èøg1 a g0 e

t ()b ++2 gd()t - t + 1 c()t 2 e 0 t d ò d 0 is valid for gt() and for any d> 0 , where b = max bt() . 0 t {}() Lemma 2.1 implies the existence of a sequence of approximate solutions um t to problem (2.1) and (2.2) on any segment []0, T .

Exercise 2.3 Show that every approximate solution um is a solution to = = ()=, problem (2.1) and (2.2) with u0 u0 m ,u1 u1 m , and hxt = (), hm xt, where m ==(), uim pmui å ui ek ek , k = 1 (2.16) m ==()(), hm pm hhtå ek ek , k = 1 { = , and pm is the orthoprojector onto the span of elements ek : k 1 ,,}¼ = 2 m in 0 H . {} Let us prove that the sequence of approximate solutions um is convergent. Auxiliary Linear Problem 227

At first we note that ml+ D ()==\ - \ 2 + ()- 2 ()\ ()2 + l2 ()2 ml, t um uml+ Aum uml+ å gk t k gk t km= + 1 for every t Î []0, T . Therefore, by virtue of Lemma 2.1 we have that

ml+ T b t 2 2 2 1 2 D ()t £ e 0 ()u , e ++l ()u , e ()ht(), e dt ml, å 1 k k 0 k 2g ò k km= + 1 0 for g >g0 . Moreover, in the case = 0 , the result of Exercise 2.2 gives that

ml+ T ()b + 1 t D () £ 0 (), 2 ++l2 (), 2 ()(), 2 ml, t e å u1 ek k u0 ek ò ht ek dt . km= + 1 0 {}\ () {}() These equations imply that the sequences um t and Aum t are the Cauchy sequences in the space C0, T ; H on any segment []0, T . Consequently, there exists a function ut() such that \ ()Î (), ; ()Î (), ; u t C 0 T H ,,ut C 0 T 1 æö\ ()- \ ()2 + ()- () = limmax èøum t u t um t ut 0 . (2.17) m ® ¥ []0, T 1 Equations (2.8) and (2.9) further imply that T T - æö\ ()+ g (), \ () + æö()+ () (), () = ò èøum t um t v t dt ò èøAum t btum t Av t dt 0 0 T = ()+ g , ()+ ()(), () u1 m u0 m v 0 ò hm t vt dt 0 () ()= () = , for all functions vt from WT such that vT 0 . Here uim , and hm t , i 01, are defined by (2.16). We use equation (2.17) to pass to the limit in this equation and to prove that the function ut() satisfies equality (2.3). Moreover, it follows from ()= () (2.17) that u 0 u0 . Therefore, the function ut is a weak solution to problem (2.1) and (2.2). In order to prove the uniqueness of weak solutions we consider the function ì ï s ï - u()t t, ts< , ()= ò d vs t í (2.18) ï t ï î 0, stT££, for s Î []0, T . Here ut() is a weak solution to problem (2.1) and (2.2) for h = 0 , = = ()Î u0 0 , and u1 0 . Evidently vs t WT . Therefore, 228 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h T T a \ \ - ()u()t + gut(), v ()t dt + ()Au() t + bt()ut(), Av ()t dt = 0 . p ò s ò s t 0 0 e r () Due to the structure of the function vs t we obtain that 4 s 1 æö us()2 + Av ()0 2 + g ut()2 dt = J ()uv, ,(2.19) 2 èøs ò s 0 where s (), = ()() (), () Js uv ò btut Avs t dt . 0 ()= ()- () £ It is evident that Avs t Avs 0 Avt 0 for ts . Therefore, æös s (), ££ç÷() () + ()× () Js uv b0 ç÷Avs 0 ò ut dt ò ut Avt 0 dt èø0 0 s s 1 2 2 2 b0 æö2 2 £ Av ()0 ++sb ut() dt ut() + Av ()0 dt . 4 s 0 ò 2 òèøt 0 0 If we substitute this estimate into equation (2.19), then it is easy to find that s ()2 + ()2 £ æö()2 + ()2 us Avs 0 CT ò èøut Avt 0 dt , 0 Î [], where s 0 T and CT is a positive constant depending on the length of the segment []0, T . This and Gronwall’s lemma imply that ut()º 0 . \ () Let us prove the energy equation. If we multiply equation (2.8) by gj t and summarize the result with respect to j , then we find that 1 æö\ \ \ \ d u 2 + Au 2 ++g u 2 bt()()Au , u = ()hu, . 2 dt èøm m m m m m After integration with respect to t we use (2.17) to pass to the limit and obtain (2.6). Theorem 2.1 is completely proved.

Exercise 2.4 Prove that the estimate æöt \ ç÷b t u()t 2 + Au() t 2 £ u 2 ++Au 2 1 h()t 2 t e 0 (2.20) ç÷1 0 2g ò d èø0 is valid for a weak solution ut() to problem (2.1) and (2.2). Here = {}() ³ g > b0 max bt : t 0 and 0 . Auxiliary Linear Problem 229

Exercise 2.5 Let ut() be a weak solution to problem (2.1) and (2.2). Prove 2 ()Î (), ; that A ut C 0 T -1 and t \ ()+ g () = + g - ()t2 ()t + ()t ()t - ()t u t ut u1 u0 ò A u b Au h d . 0 Here we treat the equality as an equality of elements in -1 . (Hint: use the results of Exercises 2.1 and 2.1.3).

Exercise 2.6 Let ut() be a weak solution to problem (2.1) and (2.2) constructed in Theorem 2.1. Then the function u\ ()t is absolutely continuous as a vector-function with the values in -1 while the derivati- \\ () ¥ (), ; ve u t belongs to the space L 0 T -1 . Moreover, the function ut() satisfies equation (2.1) if we treat it as an equality of elements Î [], in -1 for almost all t 0 T .

In particular, the result of Exercise 2.6 shows that a weak solution satisfies equation (2.1) in a stronger sense then (2.4). We also note that the assertions of Theorem 2.1 and Exercises 2.4–2.6 with the corresponding changes remain true if the initial condition is given at any other moment t0 which is not equal to zero.

Now we consider the case ht()º 0 and construct the evolutionary operator of problem (2.1) and (2.2). To do that, let us consider the family of spaces H =s ´ ³ s 1 + s s ,.0 H = () Î Î Every space s is a set of pairs yu; v such that u 1 + s and v s . We define the inner product in Hs by the formula (), = (), + (), y1 y2 u1 u2 + s v1 v2 s . Hs 1

Exercise 2.7 Prove that Hs is compactly embedded into Hs for s > s . 1 1 H (); In the space 0 we define the evolutionary operator Ut t0 of problem (2.1) and (2.2) for ht()º 0 by the equation (); = ()(); \ () Ut t0 yutu t , (2.21) where ut() is a solution to (2.1) and (2.2) at the moment t with initial conditions = (); that are equal to yu0 u1 at the moment t0 . The following assertion plays an important role in the study of asymptotic behaviour of solutions to problem (0.1) and (0.2). 230 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h Theorem 2.2 a () p Assume that the function bt is continuously differentiable in (2.1) t and such that e \ r =¥() < =¥() < b0 sup bt ,.b1 sup b t 4 t t Then the evolutionary operator Ut(); t of problem (2.1) and (2.2) for ht()º 0 is a linear bounded operator in each space Hs for s ³ 0 and it possesses the properties: a) Ut(); t U()t ; s = Ut(); s ,,,t ³³t s ,,;Ut(); t = I ; b) for all s ³ 0 the estimate æö Ut(); t y £ y exp 1 b ()t - t (2.22) Hs Hs èø2 0 is valid; g c) there exists a number N0 depending on , b0 , and b1 such that the equation g - ()t - t ()- (); t £ ()- 4 > t IPUt y H 3IPy H e ,,,t , (2.23) N s N s ³ holds for all NN0 , where PN is the orthoprojector onto the subspace = {}(); , (); = ,,¼ , LN Lin ek 0 0 ek : k 12 N H in the space 0 ...

Proof. Semigroup property a) follows from the uniqueness of a weak solution. The boundedness property of the operator Ut(); t follows from (2.22). Let us prove relations (2.22) and (2.23). It is sufficient to consider the case t = 0 . According to the definition of the evolutionary operator we have that (); = ()(); \ () = () Ut 0 yutu t ,,yu0 ; u1 where ut() is the weak solution to problem (2.1) and (2.2) for ht()º 0 . Due to (2.17) it can be represented as a convergent series of the form ¥ ()= () ut å gk t ek . k = 1 Moreover, Lemma 2.1 implies that b t \ ()2 + l2 ()2 £ ()\ ()2 + l2 ()2 0 gk t k gk t gk 0 k gk 0 e . (2.24) Since ¥ æö\ s Ut(); 0 y 2 = g ()t 2 + l2 g ()t 2 l2 , Hs å èøk k k k k = 1 equation (2.24) implies (2.22). Auxiliary Linear Problem 231

Further we use equation (2.13) to obtain that g - t \ ()2 + l2 ()2 £ æö()2 + l2 ()2 2 gk t k gk t 3 èøgk 0 k gk 0 e , (2.25) provided the conditions (cf. (2.12)) g2 \ æöl - 1 l + ££bt() 1 l ,,b()t + g k - bt() ³ 0 2 k l 2 k èø4 k are fulfilled. Evidently, these conditions hold if 4 b g2 l ³ 1 ++ k g 4 b0 , b0 = () = () where b0 max bt and b1 max bt . Since t t ¥ æö\ ()IP- Ut(); 0 y 2 = g ()t 2 + k4 g ()t 2 k2 s , N Hs å èøk k kN= + 1 ³ - equation (2.25) gives us (2.23) for all NN0 1 , where N0 is the smallest natural number such that 4 b g2 l ³ 1 ++ N g 4 b0 . (2.26) 0 b0 Thus, Theorem 2.2 is proved.

Exercise 2.8 Show that a weak solution ut() to problem (2.1) and (2.2) can be represented in the form t \ ()ut(); u()t = Ut(); 0 yUt+ ò (); t ()t0 ; h()t d , (2.27) 0 = (); (); t where yu0 u1 and Ut is defined by (2.21). Exercise 2.9 Use the result of Exercise 2.2 to show that Theorem 2.1 and Theorem 2.2 (a, b) with another constant in (2.22) also remain true for g < 0 . Use this fact to prove that if the hypotheses of Theorems 2.1 and 2.2 hold on the whole time axis, then problem (2.1) and (2.2) is solvable in the class of functions W = (); Ç 1(); C R 1 C R 0 with g ³ 0 .

Exercise 2.10 Show that the evolutionary operator Ut(), t has a bounded inverse operator in every space Hs for s ³ 0 . How is the operator - []Ut(), t 1 for t > t related to the solution to equation (2.1) for ht()º 0 ? Define the operator Ut(), t using the formula Ut()=, t - = []U()t, t 1 for t < t and prove assertion (a) of Theorem 2.2 for all t, t Î R . 232 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h § 3 Theorem on Existence a p and Uniqueness of Solutions t e r 4 In this section we use the compactness method (see, e.g., [8]) to prove the theorem on the existence and uniqueness of weak solutions to problem (0.1) and (0.2) under the assumption that Î Î ()Î ¥ (), ; u0 1 ,u1 0 ,pt L 0 T 0 ; (3.1) z ()Î 1() M ()º ()xx ³ - - Mz C R+ ,,(3.2)z òM d az b 0 £ < l Îl where 0 a 1 , b R , 1 is the first eigenvalue of the operator A , and the operator LDA is defined on () and satisfies the estimate Lu£ C Au ,.uDAÎ () (3.3) ()Î Similarly to the linear problem (see Section 2), the function ut WT is said to be a weak solution to problem (0.1) and (0.2) on the segment []0, T ()= ifu 0 u0 and the equation T T - ()\ ()+ g (), \ () + æö()+ æö12¤ ()2 (), () + ò u t ut v t dt ò èøAu t MAèøut ut Av t dt 0 0 T T + ()(), () = ()+ g , ()+ ()(), () ò Lu t vt dt u1 u0 v 0 ò pt vt dt (3.4) 0 0 ()Î ()= holds for any function vt WT such that vT 0 . Here the space WT is defined by equation (1.2).

Exercise 3.1 Prove the analogue of formula (2.4) for weak solutions to problem (0.1) and (0.2).

The following assertion holds.

Theorem 3.1 Assume that conditions (3.1)–(3.3) hold. Then on every segment []0, T problem (0.1) and (0.2) has a weak solution ut(). This solution is unique. It possesses the properties ()Î (), ; \ ()Î (), ; ut C 0 T 1 , u t C 0 T 0 (3.5) Theorem on Existence and Uniqueness of Solutions 233 and satisfies the energy equality t ()(), \ () = ()g, + æö- \ ()t 2 + ()- ()+ (), \ ()t t Eut u t Eu0 u1 ò èøu Lu t pt u d , (3.6) 0 where æöæö Eu(), v = 1 v 2 ++Au 2 M A12¤ u 2 . (3.7) 2 èøèø

We use the scheme from Section 2 to prove the theorem. The Galerkin approximate solution of the order m to problem (0.1) and (0.2) with respect to the basis ek is defined as a function of the form m ()= () um t å gk t ek k = 1 which satisfies the equations ()\\ ()+ g\ (), um t um t ej +

++æö()+ æö12¤ 2 (), ()()- (), = (3.8) èøAum t MAèøu um t Aej Lum t pt ej 0 for j = 12,,¼ ,m with t Î (0,]T and the initial conditions ()(), = (), ()\ (), = (), = ,,¼ , um 0 ej u0 ej ,um 0 ej u1 ej ,j 12 m . (3.9) Simple calculations show that the problem of determining of approximate solutions can be reduced to solving the following system of ordinary differential equations: m m \\ \ æö g ++g g l2 g +l Mç÷l g ()t 2 g +()Le , e g = p ()t , (3.10) k k k k k èøå j j k å j k j k j = 1 j = 1 ()==(), \ ()==(), = ,,¼ , gk 0 g0 k u0 ek ,gk 0 g1 k u1 ek ,k 12 m . (3.11) The nonlinear terms of this system are continuously differentiable with respect to gj . Therefore, it is solvable at least locally. The global solvability follows from the a priori estimate of a solution as in the linear problem. Let us prove this estimate. () We consider an approximate solution um t to problem (0.1) and (0.2) on the solvability interval ()0, t . It satisfies equations (3.8) and (3.9) on the interval (), \ () 0 t . We multiply equation (3.8) by gj t and summarize these equations with respect to j from 1 to m . Since d æöæö¤ 2 \ M A12¤ u 2 = 2 MA12u ()Au() t , u()t , dt èøèø we obtain \ \ \ d Eu()g()t , u ()t = - u ()t 2 - ()Lu ()t - pt(), u ()t (3.12) dt m m m m m 234 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C (), \ h as a result, where Eu u is defined by (3.7). Equation (3.3) implies that a \ \ æö2 \ 2 p ()Lu , u ££CAu u CAu + u . t m m m m èøm m e r Condition (3.2) gives us the estimate \ 4 1 æö2 + \ 2 £ C + C Eu(), u (3.13) 2 èøAum um 1 2 m m with the constants independent of m . Therefore, due to Gronwall’s lemma equation (3.12) implies that æö\ C t \ ()2 + ()2 £ + ()(), () 2 ,(3.14) um t Aum t èøC0 C1 Eum 0 um 0 e

with the constants C0 , C1 , and C2 depending on the problem parameters only.

Exercise 3.2 Use equation (3.14) to prove the global solvability of Cauchy problem (3.10) and (3.11).

It is evident that \ u ()0 £ u andA12¤ u ()0 ££1 Au ()0 1 u . m 1 m l m l 0 1 1 1 Therefore, \ 1 æö Eu()()0 , u ()0 £ u 2 ++u 2 C ()u , m m 2 èø1 0 1 M 0 1 ()r = {}M () ££r2¤ l where CM max z : 0 z 1 . Consequently, equation (3.14) gives us that æö\ ()2 + ()2 £ sup èøum t Aum t CT (3.15) t Î []0, T > for any T 0 , where CT does not depend on m . Thus, the set of approximate solu- {}() > tions um t is bounded in WT for any T 0 . Hence, there exist an element ut()ÎW and a sequence {}m such that u ()t ® ut() weakly in W . Let us T k mk T show that the weak limit point ut() possesses the property \ ()2 + ()2 £ u t Au t CT (3.16) for almost all t Î []0, T . Indeed, the weak convergence of the sequence {}u to mk \ 2 the function uW in means that um and Au weakly (in L ()0, T ; H ) con- \ T k mk verge to u and Au respectively. Consequently, this convergence will also take place in L2 ()ab, ; H for any ab and from the segment []0, T . Therefore, by virtue of the known property of the weak convergence we get b b æö\ ()2 + ()2 £ æö\ ()2 + ()2 èøu t Au t dt lim èøum t Aum t dt . ò k ® ¥ ò k k a a Theorem on Existence and Uniqueness of Solutions 235

With the help of (3.15) we find that b æö\ ()2 + ()2 £ ()- ò èøu t Au t dt CT ba. a Therefore, due to the arbitrariness of ab and we obtain estimate (3.16).

Lemma 3.1 For any function vt()Î L2()0, T ; H (), = (), lim JT um v JT uv, k ® ¥ k where T (), = æö12¤ ()2 ()(), () JT uv òMAèøut ut vt dt . 0

Proof. Since

æöæö MA12¤ u 2 - MA12¤ u 2 £ èømk èø

1 æö £ M x A12¤ u + ()1 - x A12¤ u dx × A12¤ ()u - u , ò èømk mk 0

2 where M ()z = 2 zM ()z Î C ()R+ , due to (3.15) and (3.16) we have

æöæö()1 MA12¤ u 2 - MA12¤ u 2 £ C A12¤ ()u ()t - ut() , èømk èøT mk

()1 () where the constant CT is the maximum of the function M z on the sufficient- [], ly large segment 0 aT , determined by the constant CT from inequalities (3.15) and (3.16). Hence,

T æöæö d º MA12¤ u ()t 2 - MA12¤ ut()2 × ()u ()t , vt() dt £ k ò èøm èømk 0 æöT 12¤ £ ç÷12¤ ()()- () 2 C v 2 A u t ut dt . T L ()0, T ; H ç÷ò mk èø0 2 (), ; The compactness of the embedding of WT into L 0 T 12¤ (see Exer- d ® ® ¥ cise 1.13) implies that k 0 as k . It is evident that 236 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h T a æö J ()u , v -dJ ()uv, £ + ()u ()t - ut(), vt() × MA12¤ ut()2 dt . p T mk T k ò mk èø t 0 e r Because of the weak convergence of u to u this gives us the assertion of the mk 4 lemma.

Exercise 3.3 Prove that the functional T L []u = ò()Lu() t , vt() dt 0 Î 2 (), ; is continuous on WT for any vL0 T H .

Let us prove that the limit function ut() is a weak solution to problem (0.1) and (0.2). , = ,,¼ , Let pl be the orthoprojector onto the span of elements ek k 12 l in the space H . We also assume that

= {}Î ()= WT vWT : vT 0 and

l º = {}Î WT pl WT pl v : vWT .

l It is clear that an arbitrary element of the space WT has the form l (), = h() vl xt å k t ek , k = 1 h () [], where k t is an absolutely continuous real function on 0 T such that h ()=h\ ()Î 2 (), k T 0 ,.k t L 0 T h () If we multiply equation (3.8) by j t , summarize the result with respect to j from 1 to lt , and integrate it with respect to from 0 to T , then it is easy to find that

T T T - ()\ + g , \ ++æö+ æö12¤ 2 , (), = ò um um vl dt ò èøAum MAèøum um Avl dt ò Lum vl dt 0 0 0 T = ()+ g , ()+ (), u1 u0 vl 0 ò pvl dt 0 for ml³ . The weak convergence of the sequence u to uW in as well as Lem- mk T ma 3.1 and Exercise 3.3 enables us to pass to the limit in this equality and to show Theorem on Existence and Uniqueness of Solutions 237

() Î l that the function ut satisfies equation (3.4) for any function vWT , where l = 12,,¼ . Further we use (cf. Exercise 2.1.11) the formula T æö\ ()- \ ()2 + ()- ()2 = limèøpl v t v t pl Av t Av t dt 0 l ® ¥ ò 0

()Î l for any function vt W T in order to turn from the elements vW of T to the functions vW from the space T .

Exercise 3.4 Prove that ut() = u . t = 0 0 () {} Thus, every weak limit point ut of the sequence of Galerkin approximations um in the space WT is a weak solution to problem (0.1) and (0.2). If we compare equations (3.4) and (2.3), then we find that every weak solution ut() is simultaneously a weak solution to problem (2.1) and (2.2) with bt()= 0 and

()= - æö12¤ ()2 ()- ()+ () ht MAèøut Au t Lu t pt.(3.17) ()Î ¥ (), ; It is evident that ht L 0 T 0 . Therefore, due to Theorem 2.1 equations (3.5) are valid for the function ut() . To prove energy equality (3.6) it is sufficient (due to (2.6)) to verify that for ht() of form (3.17) the equality t \ ò ()th()t , u()t d = 0 t t = t \ 1 æö¤ 2 = ()t- Lu()t + p()t , u()t d - M A12u()t (3.18) ò 2 èø t = 0 0 holds. Here ut() is a vector-function possessing property (3.5). We can do that by first proving (3.18) for the function of the form pl u and then passing to the limit.

Exercise 3.5 Let ut() be a weak solution to problem (0.1) and (0.2). Use equation (3.6) to prove that C t \ ()2 + ()2 £ ()+ (), 2 > u t Au t C0 C1Eu0 u1 e ,,t 0 (3.19) ,, > where C0 C1 C2 0 are constants depending on the parameters of problem (0.1) and (0.2).

Let us prove the uniqueness of a weak solution to problem (0.1) and (0.2). We as- () () sume that u1 t and u2 t are weak solutions to problem (0.1) and (0.2) with the {}, {}, initial conditions u01 u11 and u02 u12 , respectively. Then the function ()= ()- () ut u1 t u2 t 238 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C = - h is a weak solution to problem (2.1) and (2.2) with the initial conditions u0 u01 ¤ 2 a - u , u = u - u , the function bt()= MA()12u ()t , and the right-hand p 02 1 11 12 1 t side e r ()= æö12¤ ()2 - æö12¤ ()2 ()+ ()()- () ht MAèøu2 t MAèøu1 t Au2 t Lu2 t u1 t . 4 We use equation (3.19) to verify that () £ ()(), + (), ()()- () Î [], ht GT Eu01 u11 Eu02 u12 Au1 t u2 t ,,t 0 T ()x x where GT is a positive monotonely increasing function of the parameter . Therefore, equation (2.20) implies that \ \ u ()t - u ()t 2 + u ()t - u ()t 2 £ 1 2 1 2 1 æöt ç÷ £ C u - u 2 ++u - u 2 u ()t - u ()t 2 t , T ç÷11 12 01 02 1 ò 1 2 1 d èø0 > where CT 0 depends on T and the problem parameters and is a function of the (), = , variables Eu0 j u1 j ,j 12 . We can assume that CT is the same for all initial (), £ = , data such that Eu0 j u1 j R ,j 12 . Using Gronwall’s lemma we obtain that \ \ æöaC u ()t - u ()t 2 + u ()t - u ()t 2 £ C u - u 2 + u - u 2 e 2 ,(3.20) 1 2 1 2 1 1 èø11 12 01 02 1 where t Î []0, T and C > 0 is a constant depending only on T , the problem parameters and the value R > 0 such that u 2 + u 2 £ R . In particular, this esti- 0 j 1 1 j mate implies the uniqueness of weak solutions to problem (0.1) and (0.2). The proof of Theorem 3.1 is complete.

Exercise 3.6 Show that a weak solution ut() satisfies equation (0.1) if we consider this equation as an equality of elements in -1 for almost all t . \\ ()Î (), ; Moreover, u t C 0 T -1 (Hint: see Exercise 2.5). Exercise 3.7 Assume that the hypotheses of Theorem 3.1 hold. Let ut() be a weak solution to problem (0.1) and (0.2) on the segment []0, T () and let um t be the corresponding Galerkin approximation of the order m . Show that ()® () 2 (), ; um t ut weakly in L 0 T 1 , \ ()® \ () 2 (), ; um t u t weakly in L 0 T 0 , ()® () 2(), ; < um t ut strongly in L 0 T s , s 1, as m ® ¥ . Theorem on Existence and Uniqueness of Solutions 239

In conclusion of the section we note that in case of stationary load pt()º pHÎ we can construct an evolutionary operator St of problem (0.1) and (0.2) in the space HHº = ´ 0 1 0 supposing that = ()(); \ () St yutu t = (), () for yu0 u1 , where ut is a weak solution to problem (0.1) and (0.2) with the = (); initial conditions yu0 u1 . Due to the uniqueness of weak solutions we have = = , t ³ St ° St St + t ,,S0 I t 0 . H By virtue of (3.20) the nonlinear mapping St is a continuous mapping of . Equa- tion (3.5) implies that the vector-function St y is strongly continuous with respect to ty for any Î H . Moreover, for any R > 0 and T > 0 there exists a constant CRT(), > 0 such that - £ (), × - St y1 St y2 H CRT y1 y2 H (3.21) Î [], Î {}Î H £ for all t 0 T and for all yj y : y H R . (), ® Exercise 3.8 Use equation (3.21) to show that ty St y is a continuous mapping from R+ ´ H into H . Exercise 3.9 Prove the theorem on the existence and uniqueness of solutions to problem (0.1) and (0.2) for g £ 0 . Use this fact to show that {} the collection of operators St is defined for negative t and forms a group (Hint: cf. Exercises 2.9 and 2.10). H Exercise 3.10 Prove that the mapping St is a homeomorphism in for every t > 0 . ¥ Exercise 3.11 Let pt()Î L ()R+, H be a periodic function: pt()= = ()+ > pt t0 , t0 0 . Define the family of operators Sm by the formula = ()(); \ () = ,,,¼ Sm yumt0 u mt0 ,,m 012 H= ´ () in the space 1 0 . Here ut is a solution to problem (0.1) = (), and (0.2) with the initial conditions yu0 u1 . Prove that the ()H, = m pair Sm is a discrete dynamical system. Moreover, Sm S1 H and S1 is a homeomorphism in . 240 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h § 4 Smoothness of Solutions a p t e r In the study of smoothness properties of solutions constructed in Section 3 we use 4 some ideas presented in paper [9]. The main result of this section is the following assertion.

Theorem 4.1 Let the hypotheses of Theorem 3.1 hold. We assume that Mz()Î Î l +1 () () l (), ; ³ C R+ and the load pt lies in C 0 T 0 for some l 1 . Then for a weak solution ut() to problem (0.1) and (0.2) to possess the properties ()k ()Î (), ; = ,,,¼ ,- u t C 0 T 2 , k 012 l 1 , (4.1) ()l ()Î (), ; ()l + 1 ()Î (), ; u t C 0 T 1 , u t C 0 T 0 , it is necessary and sufficient that the following compatibility conditions are fulfilled: ()k ()Î = ,,,¼ ,- ()l ()Î u 0 2 ,,;k 012 l 1 ;;.u 0 1 . (4.2) Here u()j ()t is a strong derivative of the function ut() with respect to t of the order ju and the values ()k ()0 are recurrently defined by the initial

conditions u0 and u1 with the help of equation (0.1): ()0 ()= ()1 ()= u 0 u0 ,,,u 0 u1 , ì u()k ()0 = -í gu()k - 1 ()0 +++A2u()k - 2 ()0 Lu()k - 2 ()0 î k - 2 ü + d æöæö12¤ ()2 ()- () k - 2 èøMAèøut Au t pt ý , (4.3) dt t = 0þ where k = 23,,¼ .

Proof. It is evident that if a solution ut() possesses properties (4.1) then compatibility conditions (4.2) are fulfilled. Let us prove that conditions (4.2) are sufficient for equations (4.1) to be satisfied. We start with the case l = 1 . The compatibility condi- Î Î tions have the form: u0 2 , u1 1 . As in the proof of Theorem 3.1 we consider the Galerkin approximation m ()= () um t å gk t ek k = 1 of the order m for a solution to problem (0.1) and (0.2). It satisfies the equations Smoothness of Solutions 241

\\ \ ()++ g () 2 () + um t um t A um t

++æö12¤ ()2 () () = () MAèøum t Aum t pm Lum t pm pt , (4.4) ()= \ ()= um 0 pmu0 ,,um 0 pmu1 , ¼ where pm is the orthoprojector onto the span of elements e1 em . The structure ()Î 2 (), ; of equation (3.10) implies that um t C 0 T 2 . We differentiate equation ()= \ () (4.4) with respect to tv to obtain that m t um t satisfies the equation \\ ++g \ 2 +æö12¤ 2 + = vm vm A vm MAèøum Avm pm Lvm

= - ¢ æö12¤ 2 (), \ + \ () 2 M èøA um Aum um Aum pm p t (4.5) and the initial conditions ()= vm 0 pmu1 , ìü \ ()= -g++2 æö12¤ 2 + - () vm 0 pm íýu1 A u0 MAèøpmu Au0 Lpm u0 p 0 .(4.6) îþ0 It is clear that \ æöæö v ()0 = p u()2 ()0 ++MA12¤ u 2 - MA12¤ p u 2 Ap u Lu()- p u , m m èø0 èøm 0 m 0 0 m 0 where u()2 ()0 is defined by (4.3). Therefore, v ()0 - p u()2 ()0 £ Cu()u - p u . m m 0 1 0 m 0 1 Î ()2 ()Î The compatibility conditions give us that u0 2 and hence u 0 0 . Thus, \ () the initial condition vm 0 possesses the property \ ()- ()2 ()® ® ¥ vm 0 u 0 0 ,.m (4.7) \ () We multiply (4.5) by vm t scalarwise in H to find that æö\ \ \ 1 d v ()t 2 + Av ()t 2 + g v ()t 2 = ()F ()t , v ()t , (4.8) 2 dt èøm m m m m where

()= - æö12¤ 2 - - Fm t MAèøum Avm pm Lvm

- ¢ æö12¤ 2 (), \ + \ () 2 M èøA um Aum um Aum pm p t . (4.9) () Using a priori estimates (3.14) for um t we obtain () £ æö+ () Î [], Fm t CT èø1 Avm t ,.t 0 T 242 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h Thus, equation (4.8) implies that a d æö\ æö\ 2 p v ()t 2 + Av ()t 2 £ C 1 ++v ()t 2 Av ()t ,.t Î []0, T t dt èøm m T èøm m e r Î Equation (4.7) and the property u1 1 give us that the estimate 4 \ ()2 + ()2 < vm 0 Avm 0 C holds uniformly with respect to m . Therefore, we reason as in Section 3 and use Gronwall’s lemma to find that ()2 + \ ()2 £ Î [], Avm t vm t CT ,.t 0 T (4.10) Consequently, \ ()2 + \\ ()2 £ Î [], Aum t um t CT ,.(4.11)t 0 T Equation (4.4) gives us that 2 () £ \\ ()++ g \ () ()12¤ ()2 () +() +() A um t um t um t MA um t Aum t Lum t pt . Therefore, (3.14) and (4.11) imply that 2 () £ Î [], A um t CT ,.t 0 T (4.12) {}() Thus, the sequence um t of approximate solutions to problem (0.1) and (0.2) possesses the properties (cf. Exercise 3.7): ü u ()t ® ut() weakly in L2 ()0, T ; DA()2 ; ï m ï \ \ u ()t ® u()t weakly in L2 ()0, T ; DA(); ý (4.13) m ï \\ ()® \\ () 2(), ; ï um t u t weakly in L 0 T H ; þ

where ut() is a weak solution to problem (0.1) and (0.2). Moreover (see Exer- cise 1.13), T \ ()- \ ()2 + ()- ()2 = limum t u t um t ut + dt 0 (4.14) m ® ¥ ò s 1 s 0 for every s < 1 . If we use these equations and arguments similar to the ones given in \ Section 3, then it is easy to pass to the limit and to prove that the function wt()= u()t is a weak solution to the problem

ì \\ ++g \ 2 +æö12¤ ()2 + = ï w w A wMAèøut Aw Lw ï ï = - ¢ æö12¤ ()2 (), \ + \ () í 2 M èøA ut Au u Au p t , (4.15) ï ï ï ()= \ ()= ()2 () î w 0 u1 , w 0 u 0 , Smoothness of Solutions 243 where u()2 ()0 is defined by (4.3). Therefore, Theorem 2.1 gives us that \ u()t = wt()Î C()0, T ; DA()Ç C1 ()0, T ; H . This implies equation (4.1) for l = 1 .

Further arguments are based on the following assertion.

Lemma 4.1 Let ut() be a weak solution to the linear problem ì \\ \ ï u()t ++ gu()t A2 ut() +bt()Au = ht() , í (4.16) ï u()0 = u , î 0 where bt() is a scalar continuously differentiable function, ht()Î Î 1 (), ; Î Î C 0 T 0 and u0 2 , u1 1 . Then ()Î (), ; ÇÇ1 (), ; 2 (), ; ut C 0 T 2 C 0 T 1 C 0 T 0 (4.17) \ and the function vt()= u()t is a weak solution to the problem obtained by the formal differentiation of (4.16) with respect to t and equiped ()= \()= \\ ()º -( g ++2 with the initial conditions v 0 u1 and v 0 u 0 u1 A u0 +)() - () b 0 Au0 h 0 .

Proof. () Let um t be the Galerkin approximation of the order m of a solution to () problem (4.16) (see (2.7)). It is clear that um t is thrice differentiable with re- ()= \ () spect to tv and m t um t satisfies the equation \ \\ ++g \ 2 +() = - () + () > vm vm A vm bt Avm b t Aum pmht ,,t 0 and the initial conditions ()= \ ()= -g()++2 () - () vm 0 pm u1 ,.vm 0 pm u1 A u0 b 0 Au0 h 0 Therefore, as above, it is easy to prove the validity of equations (4.10)–(4.14) for the case under consideration and complete the proof of Lemma 4.1.

Exercise 4.1 Assume that the hypotheses of Lemma 4.1 hold with bt()Î Î l (), ()Î l (), ; ³ C 0 T and ht C 0 T 0 for some l 1 . Let the com- ()0 ()= ()1 ()= patibility conditions (4.2) be fulfilled with u 0 u0 , u 0 = u1 , and ì ()- ()- u()k ()0 = -íg u k 1 ()0 ++A2 u k 2 ()0 î k - 2 ++j ()j () ()k - 2 - j () å Ck - 2 b 0 Au 0 j = 0 ü + Lu()k - 2 ()0 - h()k - 2 ()0 ý þ 244 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C = ,,¼ , () h for k 23 l . Show that the weak solution ut to problem () a (4.16) possesses properties (4.1) and the function v ()t = u k ()t is p k t a weak solution to the equation obtained by the formal differentia- e tion of (4.16) ktktimes with respect to . Here = 01,,¼ ,l r .

4 In order to complete the proof of Theorem 4.1 we use induction with respect to l . Assume that the hypotheses of the theorem as well as equations (4.2) for ln= +1 hold. Assume that the assertion of the theorem is valid for ln= ³ 1 . Since equations (4.1) hold for the solution ut() with ln= , we have

k ìü d æö12¤ ()2 () = æö12¤ ()2 ()k ()+ () íýMAèøut Au t MAèøut Au t Gk t , dtk îþ ()Î 1 (), ; = ,,¼ where Gk t C 0 T 0 , k 1 n . Therefore, we differentiate equation (0.1) n -1 times with respect to tvt to obtain that ()= u()n -1 ()t is a weak solution to problem (4.16) with

()= æö12¤ ()2 ()= - ()+ ()n -1 () bt MAèøut andht Gn -1 t p t . \ Consequently, Lemma 4.1 gives us that wt()= v()t is a weak solution to the problem which is obtained by the formal differentiation of equation (0.1) n times with respect to t : ì \\ ++g \ 2 +æö12¤ ()2 = ()n ()- () ï w w A wMAèøut Aw p t Gn t , í ï ()n \ ()n +1 î w()0 = u ()0 , w()0 = u ()0 . However, the hypotheses of Lemma 4.1 hold for this problem. Therefore (see (4.17)), ()n () = ()Î (), ; ÇÇ1 (), ; 2 (), ; u t wt C 0 T 2 C 0 T 1 C 0 T 0 , i.e. equations (4.1) hold for ln= +1 . Theorem 4.1 is proved.

Exercise 4.2 Show that if the hypotheses of Theorem 4.1 hold, then the function vt()= u()k ()t is a weak solution to the problem which is obtained by the formal differentiation of equation (0.1) k times with respect to tk , = 12,,¼ ,l .

Exercise 4.3 Assume that the hypotheses of Theorem 4.1 hold and L º 0 in equation (0.1). Show that if the conditions ()Î k ()[], ; = ,,¼ , pt C 0 T lk- ,,k 01 l (4.18) are fulfilled, then a solution ut() to problem (0.1) and (0.2) possesses the properties ()Î k ()[], ; = ,,¼ ,+ ut C 0 T l +1 - k ,.k 01 l 1 Smoothness of Solutions 245

Exercise 4.4 Assume that the hypotheses of Theorem 4.1 hold. We define the sets ì ü V = í()u , u Î H : equation (4.2) holds with lk= ý (4.19) k î 0 1 þ H= ´ in the space 1 0 . Prove that V = ´ V ÉÉÉV ¼ V 1 2 1 and1 2 l . V Exercise 4.5 Show that every set k given by equality (4.19) is invariant: (); Î V Þ ()(); \ () Î V = ,,¼ u0 u1 k ut u t k ,.k 1 l Here ut() is a weak solution to problem (0.1) and (0.2).

Exercise 4.6 Assume that L º 0 in equation (0.1) and the load pt() pos- = ,,¼ , V sesses property (4.18). Show that for k 12 l the set k of ´ form (4.19) contains the subspace k + 1 k . Exercise 4.7 Assume that the hypotheses of Theorem 3.1 hold and the operator L (in equation (0.1)) possesses the property £ << Lu s Cu1 + s for some0 s 1 . (4.20) Î Î Let u0 1 + s and let u1 s . Show that the estimate \ u ()t 2 + u ()t 2 £ C ,,(4.21)t Î []0, T m s m 1 + s T () is valid for the approximate Galerkin solution um t to problem (0.1) and (0.2). Here the constant CT does not depend on m l2 s \ () (Hint: multiply equation (3.8) by j gj t and summarize the result with respect to j ; then use relation (3.14) to estimate the nonlinear term).

Exercise 4.8 Show that if the hypotheses of Exercise 4.7 hold, then problem (0.1) and (0.2) possesses a weak solution ut() such that ()Î (), ; ÇÇ1(), ; 2(), ; ut C 0 T 1+ s C 0 T s C 0 T - 1 + s , where s Î ()01, is the number from Exercise 4.7. 246 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h § 5 Dissipativity and Asymptotic a p Compactness t e r 4 In this section we prove the dissipativity and asymptotic compactness of the dynam- ()H, ical system St generated by weak solutions to problem (0.1) and (0.2) for g > ()º Î = H = 0 in the case of a stationary load pt pH 0 . The phase space is = ´ 1 0 . The evolutionary operator is defined by the formula = ()(), \ () St yutu t , (5.1) where ut() is a weak solution to problem (0.1) and (0.2) with the initial condition = (); yu0 u1 .

Theorem 5.1 Assume that in addition to (3.2) the following conditions are fulfilledfulfilled:: > a) there exist numbers aj 0 such that z ()- ()xx ³ 1 + a - ³ zM z a1 òM d a2 z a3 , z 0 (5.2) 0 with a constant a > 0 ; b) there exist 0 £ q < 1 and C > 0 such that Lu£ C Aq u ,,.uDAÎ ()q . (5.3) ()H, Then the dynamical system St generated by problem (0.1) and (0.2) for g > 0 and for pt()º pHÎ is dissipativedissipative..

To prove the theorem it is sufficient to verify (see Theorem 1.4.1 and Exercise 1.4.1) that there exists a functional Vy() on H which is bounded on the bounded sets of the space H , differentiable along the trajectories of system (0.1) and (0.2), and such that ()³ a 2 - D Vy y H 1 , (5.4)

dVS()y + b VS()y £ D ,(5.5) dt t t 2 ab, > D , D ³ () where 0 and 1 2 0 are constants. To construct the functional Vy we use the method which is widely-applied for finite-dimensional systems (we used it in the proof of estimate (2.13)). Let Vy()= Ey()+ nB()y , = (); Î H ()= (); where yu0 u1 . Here Ey Eu0 u1 is the energy of system (0.1) and (0.2) defined by the formula (3.7), Dissipativity and Asymptotic Compactness 247

g B()y = ()u ; u + u 2 , 0 1 2 0 and the parameter n > 0 will be chosen below. It is evident that

-B1 u 2 ££()y 1 u 2 + g u 2 . 2g 1 2g 1 0 For 0 <n with the constants b1 b2 0 independent of . This inequality guarantees the boundedness of Vy() on the bounded sets of the space H . ()() ()= Energy equality (3.6) implies that the function Eyt , where yt St y , is continuously differentiable and \ \ d Eyt()() = - gu()t 2 + ()-Lut()+ p, u()t . dt Therefore, due to (5.3) we have that g \ d Eyt()() £ - u 2 ++C Aqu 2 C . dt 2 1 2 We use interpolation inequalities (see Exercises 2.1.12 and 2.1.13) to obtain that q 2 2 12¤ 2 A u £ d Au + Cd A u ,.d > 0 Thus, the estimate g e d Eyt()() £ - u\ 2 + Au 2 ++C A12¤ u 2 C (5.7) dt 2 2 e 2 holds for any e > 0 .

Lemma 5.1 Let ut() be a weak solution to problem (0.1) and (0.2) and let yt()= =B()ut(), u\ ()t . Then the function ()yt() is continuously differentiable and \ \\ \ d B()yt() = u()t 2 + ()u()t + gu()t , ut() . (5.8) dt

\\ ()Î (), We note that since u t C R+ -1 (see Exercise 3.6), equation (5.8) is correct- ly defined.

Proof. It is sufficient to verify that t ìü ()(), \ () = (), + ()\\ ()t , ()t + \ ()t 2 t ut u t u0 u1 ò íýu u u d . (5.9) îþ 0 248 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C {}, = ,,¼ , h Let pl be the orthoprojector onto the span of elements ek k 12 l in a . Then it is evident that the vector-function u ()t = p ut() is twice continu- p 0 l l t ously differentiable with respect to t . Therefore, e r t \ \ ìü\\ \ ()u ()t , u ()t = ()u ()0 , u ()0 + íý()u ()t , u ()t + u ()t 2 dt . 4 l l l l ò îþl l l 0

The properties of the projector pl (see Exercise 2.1.11) enable us to pass to the limit l ® ¥ and to obtain (5.9). Lemma 5.1 is proved.

Since ut() is a solution to equation (0.1), relation (5.8) implies that ìü d B()() = \ 2 - 2 ++æö12¤ 2 × 12¤ 2 ()- , yt u íýAu MAèøu A u Lu p u . dt îþ Therefore, equation (5.2) and the evident inequality

()Lu- p, u £ 1 Au 2 ++C u 2 p 2 2 1 give us that \ æö + a d B()yt() £ u 2 - 1 Au 2 - a M A12¤ u 2 - a A12¤ u 22 ++C u 2 C . dt 2 1 èø2 1 2 Hence, (5.6) and (5.7) enable us to obtain the estimate \ d Vyt()d() + Vyt()() £--1 ()gd- b - 2 n u 2 dt 2 1 æödb æö - 1 ()nd-eb - Au 2 - n a - 1 M A12¤ u 2 + Ru(); ned,, , 2 1 èø1 2 èø where d > 0 and ()n; ned,, = - 12¤ 22+ a +++12¤ 2 n 2 Ru a2 A u Ce A u C1 u Cd . Therefore, for any 0 <

Exercise 5.1 Prove that if the hypotheses of Theorem 5.1 hold, then the assertion on the dissipativity of solutions to problem (0.1) and (0.2) ¥ remains true in the case of a nonstationary load pt()Î L ()R+, H .

Theorem 5.2 Let the hypotheses of Theorem 5.1 hold and assume that for some s > 0 s s p Î s ,,,LD() A Ì DA(), A Lu£ C Au . (5.10)

Then there exists a positively invariant bounded set Ks in the space Hs = = ´ H 1+ s s which is closed in and such that Dissipativity and Asymptotic Compactness 249

g ìü- ()tt- 4 B sup íýdist()S yK, s : yBÎ £ Ce (5.11) îþH t H > for any bounded set B in the space , ttB .

Due to the compactness of the embedding of Hs in H for s > 0 , this theorem and ()H, Lemma 1.4.1 imply that the dynamical system St is asymptotically compact.

Proof. ()H, > Since the system St is dissipative, there exists R 0 such that for all Î ³ = () yB and tt0 t0 B \ u()t 2 + Au() t 2 £ R2 , (5.12) where ut() is a weak solution to problem (0.1) and (0.2) with the initial conditions = (); Î () yu0 u1 B . We consider ut as a solution to linear problem (2.1) and (2.2) with bt()= MA()12¤ ut()2 and ht()= - Lu() t + p . It is easy to verify that bt() is a continuously differentiable function and \ ()+ () £ ³ bt b t CR ,.tt0 Moreover, equation (2.27) implies that = (), ()+ (), ; St yUtt0 yt0 Gt t0 y , (5.13) where t (), ; = - (), t ()t; ()t - Gtt0 y ò Ut 0 Lu p d .

t0 Here Ut(), t is the evolutionary operator of the homogeneous problem (2.1) and (2.2) with ht()= 0 and bt()= MA()12¤ ut()2 . By virtue of Theorem 2.2 there ³ existsN0 0 such that ìüg ()- (), t £ - ()- t 1 PN Ut h H 3 h H exp íýt , (5.14) s s îþ4 ³ ³³t H where NN0 , t t0 , and PN is the orthoprojector onto L = {(); , (); =},,¼ , N Lin ek 0 0 ek ,.k 12 N This implies that t ìüg ()1 - P Gt(), t; y £ 3 exp - ()t - t Lu()t - p dt . N 0 H ò íý s 0 s îþ4 t0 Therefore, we use (5.10) to obtain that ()- (), ; £ () 1 PN Gt t0 y CR. (5.15) 0 Hs 250 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h It is also easy to find that a s s p P S y ££N S y RN ,.tt³ N t H 0 t H 0 0 t 0 s e r Consequently, there exists a number Rs depending on the radius of dissipativity R 4 and the parameters of the problem such that the value S y - ()1 - P Ut(), t S y = P S y + ()1 - P Gt(), t; y (5.16) t N0 0 t0 0 N0 t N0 0 lies in the ball B = {}y : y £ R s Hs s ³ for tt0 . Therefore, with the help of (5.12) and (2.23) we have that g - ()tt- 4 0 distH ()S yB, s ££()1 - P Ut(), t S yR3 e . (5.17) t N0 0 t0 = g+()º () Let Ks Bs Èt ³ 0 St Bs . Evidently equation (5.11) is valid. Moreover, Ks is positively invariant. The continuity of St y with recpect to the both variables ()ty; in the space H (see Exercise 3.8) and attraction property (5.17) imply that Ks is a closed set in H . Let us prove that Ks is bounded in Hs . First we note that H £ Ks is bounded in . Indeed, by virtue of the dissipativity we have that St yR Î ³ º () for all yBs and tts tBs . Since St y is continuous with respect to the variables ()ty; , its maximum is attained on the compact []0, ts ´ Bs . Thus, there exists > £ Î = R 0 such that yR for all yKs . Let us return to equality (5.16) for t0 0 Î H and y0 Bs . It is evident that the norm of the right-hand side in the space s is bounded by the constant CC= ()s, R . However, equation (5.14) implies that ()- (), £ ³ Î 1 PN Ut 0 y0 3 Rs ,,t 0 y0 Bs . 0 Hs Therefore, equation (5.16) leads to the uniform estimate

S y £ Rs ,,t ³ 0 yBÎ s . t Hs

Thus, the set Ks is bounded in Hs . Theorem 5.2 is proved.

££s H Exercise 5.2 Show that for any 0 s a bounded set of s is attracted by Ks at an exponential rate with respect to the metric of the space H . Thus, we can replace distH by distH in (5.11). s s Exercise 5.3 Prove that if the hypotheses of Theorem 5.2 hold, then the assertion on the asymptotic compactness of solutions to problem (0.1) and (0.2) remains true in the case of nonstationary load pt()Î ¥ Î L ()R+ , s (see also Exercise 5.1). Exercise 5.4 Prove that the hypotheses of Theorem 5.1 and 5.2 hold for problem (0.3) and (0.4) for any 0 < ~~0 (), º () 1()W and px t px lies in the Sobolev space H0 . Dissipativity and Asymptotic Compactness 251~~

~~Let us consider the dissipativity properties of smooth solutions (see Section 4) to problem (0.1) and (0.2).~~

Theorem 5.3 l + 1 Let the hypotheses of Theorem 5.1 hold. Assume that Mz()Î C ()R+ = (); and the initial conditions y0 u0 u1 are such that equations (4.1) (and ()(); \ () = hence (4.2)) are valid for the solution ut u t St y0 . Then there exists > = (); Rl 0 such that for any initial data y0 u0 u1 possessing the property u()k +1 ()0 2 ++Au()k ()0 2 A2 u()k -1 ()0 2 < r2 ,,,k = 12,,¼ ,l , (5.18) ()(); \ () = the solution ut u t St y0 admits the estimate ()k +1 ()2 ++()k ()2 2 ()k -1 ()2 < 2 u t Au t A u t Rl (5.19) = ,,¼ , ³ ()r for all k 12 l as soon as tt0 .

~~We use induction to prove the theorem. The proof is based on the following assertion.~~

~~Lemma 5.2 Assume that the hypotheses of Theorem 5.3 hold for l = 1 . Then the dy- ()H , namical system 1 St generated by problem (0.1) and (0.2) in the H = ´ space 1 2 1 is dissipative.~~

Proof. ()(); \ () = Let ut u t St y0 be a semitrajectory of the dynamical system ()H , S and let y = ()u ; u Î H = ´ . If the hypotheses of the lem- 1 t 0 0 1 \ 1 2 1 ma hold, then the function wt()= u()t is a weak solution to problem (4.15) obtained by formal differentiation of (0.1) with respect to t (as we have shown in Section 4). By virtue of Theorem 2.1 the energy equality of the form (2.6) holds for the function wt() . We rewrite it in the differential form: \ \ d Ft()g; wt(), w()t +Yw()t 2 = ()ut(), wt() , (5.20) dt where \ æö\ æö Ft(); w, w = 1 w 2 ++Aw 2 MA12¤ ut()2 × A12¤ w 2 (5.21) 2 èøèø and \ Y()ut(), wt() =+-()Lw() t , w()t

+ ¢æö12¤ ()2 ()(), \ () 12¤ ()2 - ()(), \ () M èøA ut Au t u t A wt 2 Au t w t . ()H, The dissipativity property of St given by Theorem 5.1 leads to the estimates 252 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h æö12¤ ()2 12¤ 2 £ () a MAèøut A w CR Aw t , p t \ \ e Y()ut(), wt() £ Lw w + C ()Aw() t + w()t r R for all ()u ; u with the property 4 0 1 2 +r2 £ 2 Au0 u1 ³ ()r and for all tt0 large enough. Hereinafter R is the radius of dissipativity ()H, ³ ()r of the system St . These estimates imply that for tt0 we have \ æö\ 1 æö\ 2 + 2 - a ££Ft()a; w, w wt()2 + w()t 2 + a (5.22) 4 èøw Aw 1 2 èø3 and \ g \ d Ft(); w, w +b w 2 £ Aq w 2 ++a Aw a ,(5.23) dt 2 4 5 a > q < where the constants j 0 depend on R . Here 1 . Similarly, we use Lem- ma 5.1 to find that ìüg d ()(), \ () + ()2 £ \ ()2 - 1 ()2 + íýwt w t wt w t Aw t CR dt îþ2 2 ³ ()r for tt0 . Consequently, the function \ ìü\ g Vt()= Ft()n; w, w + íý()ww, + w 2 îþ2 possesses the properties V d + w V £wC ,,> 0 dt R and æö\ æö\ 1 w 2 + Aw 2 - a ££Vaw 2 + w 2 + a 4 èø1 2 èø3 ³n()r > for tt0 and for 0 small enough. This implies that \ ()2 + ()2 £ æö\ ()2 + ()2 -w t + w t Aw t C1 èøw t0 Aw t0 e C2 (5.24) ³ = ()r for tt0 t0 , provided that 2 +r2 £ 2 Au0 u1 . (5.25) If we use (5.4)–(5.6), then it is easy to find that ()2 + \ ()2 £ > Au t u t Cr , R ,,t 0 under condition (5.25). Using the energy equality for the weak solutions to problem (4.15) we conclude that Dissipativity and Asymptotic Compactness 253

~~æö\ \ d Aw() t 2 + w()t 2 £ C ()r, R ××Aw w + C ()r, R . dt èø1 2 Therefore, standard reasoning in which Gronwall’s lemma is used leads to~~

()2 + \ ()2 £ æö()2 ++\ ()2 bt Aw t w t èøAw 0 w 0 a e , provided that equation (5.25) is valid. Here ab and are some positive constants depending on r and R . This and equation (5.24) imply that

~~\ ()2 + ()2 £ ()r, æö()+ \ ()2 + ()2 -w t + w t Aw t C1 R èø1 w 0 Aw 0 e C2 ,(5.26)~~

where C2 depends on R only. Since \ r 2 w()0 2 + Aw()0 2 £ C , u 2 + Au 2 £ æö r 1 0 èøl 1 provided that 2 +r2 2 £ 2 Au1 A u0 , equation (5.26) gives us the estimate \\ ()2 + \ ()2 £ 2 ³ ()r u t Au t bR ,.tt0 ()H , This easily implies the dissipativity of the dynamical system 1 St . Thus, Lemma 5.2 is proved.

()H , Exercise 5.5 Prove that the dynamical system 1 St generated by = (); Î H = problem (0.1) and (0.2) with the initial data y0 u0 u1 1 = ´ 2 1 is asymptotically compact provided that equations (5.10) hold.

In order to complete the proof of Theorem 5.3, we should note first that Lemma 5.2 coincides with the assertion of the theorem for l = 1 and second we should use the fact that the derivatives u()k ()t are weak solutions to the problem obtained by differentiation of the original equation. The main steps of the reasoning are given in the following exercises.

Exercise 5.6 Assume that the hypotheses of Theorem 5.3 hold for ln= +1 and its assertion is valid for ln= . Show that wt()= u()n +1 ()t is a weak solution to the problem of the form ì \\ ()++ g\ () 2 () +12¤ 2 += () ï w t w t A wt MA u Aw Lw Gn +1 t , í ï ()= ()n +1 (), \ ()= ()n + 2 () î w 0 u 0 w 0 u 0 , where () £ () > ()r Gn +1 t CRn for alltt0 . 254 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h Exercise 5.7 Use the result of Exercise 5.6 and the method given in the a ()= ()n +1 () p proof of Lemma 5.2 to prove that wt u t can be estima- t ted as follows: e r \ w()t 2 + Aw() t 2 £ 4 £ æö\ ()2 ++()2 -w t + (5.27) C1èøw 0 Aw 0 C2 e C3 , w > r = , where 0 , the numbers Cj depend on and Rn , j 12 , and the constant C3 depends on Rn only. Exercise 5.8 Use the induction assumption and equation (5.27) to prove the assertion of Theorem 5.3 for ln= +1 .

~~§ 6 Global Attractor and Inertial Sets~~

The above given properties of the evolutionary operator St generated by problem (0.1) and (0.2) in the case of stationary load pt()º p enable us to apply the general assertions proved in Chapter 1 (see also [10]).

Theorem 6.1 Assume that conditions (3.2), (5.2), (5.3), and (5.10) are fulfilled. Then ()H, the dynamical system St generated by problem (0.1) and (0.2) possesses a global attractor A of a finite fractal dimension. This attractor is HH= ´ a connected compact set in and is bounded in the space s 1+ s s , where s > 0 is defined by condition (5.10).

~~Proof. By virtue of Theorems 5.1, 5.2, and 1.5.1 we should prove only the finite dimensionality of the attractor. The corresponding reasoning is based on Theorem 1.8.1 and the following assertions.~~

Lemma 6.1. Assume that conditions (3.2), (5.2), and (5.3) are fulfilled. Let pHÎ . {}³ = , Then for any pair of semitrajectories St yj : t 0 , j 12, posses- £ ³ sing the property St yj R for all t 0 the estimate - £ ()- ³ St y1 St y2 H exp a0 t y1 y2 H ,,t 0 (6.1)

~~holds with the constant a0 depending on R . Global Attractor and Inertial Sets 255~~

Proof. () () If u1 t and u2 t are solutions to problem (0.1) and (0.2) with the initial = (); = (); ()= conditions y1 u01 u11 and y2 u02 u12 , then the function vt = ()- () u1 t u2 t satisfies the equation \\ ++g \ 2 = (),, v v A v u1 u2 t , (6.2) where (),,= æö12¤ ()2 ()- æö12¤ ()2 ()- () u1 u2 t MAèøu2 t Au2 t MAèøu1 t Au1 t Lv t . It is evident that the estimate (),, £ × ()()- () u1 u2 t CR Au1 t u2 t \ holds, provided that y ()t 2 = u ()t 2 + u ()t 2 £ R2 Therefore, (2.20) i H i i 1 . implies that t - 2 £ - 2 + - 2 t St y1 St y2 H y1 y2 H a1 ò St y1 St y2 H d . 0 Gronwall’s lemma gives us equation (6.1).

~~Lemma 6.2~~

Assume that the hypotheses of Theorem 6.1 hold. Let Ks be the compact positively invariant set constructed in Theorem 5.2. Then for any , Î y1 y2 Ks the inequality g - t æöLs a t Q ()S y - S y £ a e 4 ç÷1 + e 3 y - y (6.3) N t 1 t 2 H 2 ls 1 2 H èøN +1 = - ³ is valid, where QN 1 PN , NN0 , the orthoprojector PN and the number N0 are defined as in (5.14), Ls and ai are positive constants which depend on the parameters of the problem.

Proof. It is evident that = ()(); \ () QN St yi qN ui t qN ui t , { where qN is the orthoprojector onto the closure of the span of elements ek : =}+ ,,+¼ = ()= ()()- () kN1 N 2 in 0 H . Moreover, the function wN t qN u1 t u2 t is a solution to the equation

~~\\ ++g \ 2 +æö12¤ 2 = B(),, wN wN A wN MAèøu1 AwN N u1 u2 t , where~~

~~B (),,= - æö12¤ 2 - æö12¤ 2 - ()- N u1 u2 t MAèøu1 MAèøu2 qN Au2 qN Lu1 u2 . 256 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow~~

C Let us estimate the value B . Since y H £ Rs for yKÎ (see the proof h N s s a of Theorem 5.2), we have p -s -s t q Au =lq u ££q u l R . e N 2 N 2 1 N +1 N 2 1 + s N +1 s r Using equation (5.10) we similarly obtain that 4 ££l-s s l-s qN Lv N +1 A Lv C N +1 Av . Therefore, ()()- () £ l-s - qN Lu1 t u2 t C N +1 St y1 St y2 H . Consequently,

CR()s B ()u ,,u t £ S y - S y . (6.4) N 1 2 ls t 1 t 2 H N +1 Using equation (2.27) we obtain that t ()- = (), ()- + (), t ()t; B ()t QN St y1 St y2 Ut 0 QN y1 y2 òUt 0 N d , 0 where Ut(), t is the evolutionary operator of homogeneous problem (2.1) with ()= ()12¤ ()2 ()= bt MA u1 t and ht 0 . Therefore, (2.23) and (6.4) imply that ()- £ QN St y1 St y2 H

~~g ìüt g - t ïïCR()s t £ 3 e 4 íýy - y + e4 S y -tS y d . (6.5) 1 2 H ls ò t 1 t 2 ïïN +1 îþ0 We substitute (6.1) in this equation to obtain (6.3). Lemma 6.2 is proved.~~

³ Let us choose t0 and NN0 such that ìüg d -s a exp íý- t = ,,.Ls l + exp{}a t £d 1 < 1 2 îþ4 0 2 N 1 3 0 Then Lemmata 6.1 and 6.2 enable us to state that - £ - S y S y ly y H t0 1 t0 2 H 1 2 and ()- £ d - Q S y S y y y H , N t0 1 t0 2 H 1 2 a t = 0 0 A where le and the elements y1 and y2 lie in the global attractor . Hence, we can use Theorem 1.8.1 with M = A , VS= , and PP= . Therefore, the fractal t0 N dimension of the attractor A is finite. Thus, Theorem 6.1 is proved. Global Attractor and Inertial Sets 257

Theorem 5.2 and Lemmata 6.1 and 6.2 enable us to use Theorem 1.9.2 to obtain an assertion on the existence of the inertial set (fractal exponential attractor) for the ()H, dynamical system St generated by problem (0.1) and (0.2).

~~Theorem 6.2 Assume that the hypotheses of Theorem 6.1 hold. Then there exists A Ì H a compact positively invariant set exp of the finite fractal dimension such that~~

~~ìü-n ()tt- (), A Î £ B sup íýdistH St y exp : yB Ce îþ H ³ n for any bounded set B in and ttB . Here C and are positive num- A H bers. The inertial set exp is bounded in the space s .~~

~~To prove the theorem we should only note that relations (5.11), (6.1), and (6.3) coincide with conditions (9.12)–(9.14) of Theorem 1.9.2.~~

Using (1.8.3) and (1.9.18) we can obtain estimates (involving the parameters of the problem) for the dimensions of the attractor and the inertial set by an accurate ob- serving of the constants in the proof of Theorems 5.1 and 5.2 and Lemmata 6.1 and 6.2. However, as far as problem (0.3) and (0.4) is concerned, it is rather difficult to evaluate these estimates for the values of parameters that are very interesting from the point of view of applications. Moreover, these estimates appear to be quite overstated. Therefore, the assertions on the finite dimensionality of an attractor and inertial set should be considered as qualitative results in this case. In particular, this assertions mean that the nonlinear flutter of a plate is an essentially finite-dimensional phenomenon. The study of oscillations caused by the flutter can be reduced to the study of the structure of the global attractor of the system and the properties of inertial sets.

~~Exercise 6.1 Prove that the global attractor of the dynamical system generated by problem (0.1) and (0.2) is a uniformly asymptotically stable set (Hint: see Theorem 1.7.1).~~

We note that theorems analogous to Theorems 6.1 and 6.2 also hold for a class of retarded perturbations of problem (0.1) and (0.2). For example, instead of (0.1) and (0.2) we can consider (cf. [11–13]) the following problem

~~\\ ++g \ 2 +æö12 2 + +()= u u A uMAèøu Au Lu q ut p , \ u = u , u = u , u = j ()t . t = 0 0 t = 0 1 trÎ ()- , 0 258 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow~~

C h Here A , ML , and are the same as in Theorems 6.1 and 6.2, the symbol ut denotes a the function on []-r, 0 which is given by the equality u ()s =sÎut()+ s for p t t Î []-r, 0 , the parameter rq is a delay value, and (). is a linear mapping from e L2()-r, 0 ; into H possessing the property r 1 + a 4 0 a ()2 £ 1 + a ()s 2 s A qv c0 ò A v d -r a Îa[], a for c0 small enough and for all 0 0 , where 0 is a positive number. Such a formulation of the problem corresponds to the case when we use the model of the linearized potential gas flow (see [11–14]) to take into account the aerodynamic pressure in problem (0.3) and (0.4).

~~The following assertion gives the time smoothness of trajectories lying in the attractor of problem (0.1) and (0.2).~~

Theorem 6.3 Assume that conditions (3.2) and (5.2) are fulfilled and the linear operator L possesses the property As Lu£ C As + 12¤ u , uDAÎ () (6.6) s Î [- ¤ ,)¤ Î for all 12 12 . Let p 12¤ . Then the assertions of Theorem 6.1 l + 1 are valid for any s Î ()012, ¤ . Moreover, if Mz()Î C ()R+ for some \ l ³ 1 , then the trajectories yut= ()(), u()t lying in the global attractor A ()H, g> of the system St generated by problem (0.1) and (0.2) for 0 possess the property ()k + 1 ()2 ++()k ()2 2 ()k -1 ()2 £ 2 u t Au t A u t Rl (6.7) -¥ <<¥ = ,,¼ , for all t , k 12 l , where Rl is a constant depending on the problem parameters onlyonly..

Proof. It is evident that conditions (5.3) and (5.10) follow from (6.6). Therefore, we can apply Theorem 6.1 which guarantees the existence of a global attractor A . Let l + 1 \ us assume that Mz()Î C ()R+ , l ³ 1 . Let yt()= ()ut(), u()t be a trajectory in A -¥ <<¥ ()= () , t . We consider a function um t pm ut , where pm is the ortho- {},,¼ projector onto the span of the basis vectors e1 em in 0 for m large enough. ()Î 2 (); It is clear that um t C R 2 and satisfies the equation \\ ++g \ 2 +æö12¤ ()2 + = um um A um MAèøut Aum pm Lu pm p .(6.8)

Equation (6.6) for s = -12¤ implies that pm Lu is a continuously differentiable function. It is also evident that MA()12¤ ut()2 Î C1()R . Therefore, we differentiate equation (6.8) with respect to t to obtain the equation Global Attractor and Inertial Sets 259

~~\\ ++g \ 2 +æö12¤ ()2 = - \ ()+ () wm wm A wm MAèøut Awm pm Lu t Fm t ()==\ () \ () for the function wm t um t pmu t . Here~~

()= - ¢ æö12¤ ()2 ()(), \ () () Fm t 2 M èøA ut Au t u t Aum t . \ Since any trajectory yut= ()(), u()t lying in the attractor possesses the property \ ()2 + ()2 £ 2 -¥ <<¥ u t Au t R0 ,,t (6.9) it is clear that F ()t £ C ,.-¥ <

Therefore, as in the proof of Theorem 2.2, we find that there exists N0 such that g - ()t - t ()- (), t £ ()- 4 1 PN Ut y H C 1 PN y H e (6.12) S S H = ´ for all real s , where s 1 + s s , PN is the orthoprojector onto = {}(), ; (), = ,,¼ , LN Lin ek 0 0 ek : k 12 N , ³ (), t NN0 , and Ut is the evolutionary operator of the problem \\ \ ì u ++g u A2 ubt +()Au = 0 , í \ u = u , u = u , î t = 0 0 t = 0 1 () ()= ()(), \ () with bt of the form (6.10). Moreover, zm t wm t wm t can be presented in the form t ()= (), ()+ (), t ()t; - \ ()t + ()t zm t Ut t0 zm t0 ò Ut 0 pm Lu Fm d . (6.13)

~~t0 >> Then for mNN0 we have g - ()tt- ()- () £ 4 0 () + 1 PN zm t H Ce zm t0 H -12¤ -12¤~~

~~t g - ()t - t \ + Ce4 p Lu()t + F ()t t . ò m m -12¤ d~~

~~t0 260 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow~~

C s = - ¤ h Therefore, we use (6.9), (6.10), and (6.6) for 12 to obtain that a g t g p - ()tt- - ()t - t t ()1 - P z ()t £ C e 4 0 + C e 4 dt . e N m H R m R ò -12¤ 01 0 r t0 4 ® -¥ We tend t0 in this inequality to find that ()- () £ sup 1 PN zm t H C , tRÎ -12¤ where C > 0 does not depend on m . It further follows from (6.9) and equation (0.1) that () £ sup PN zm t H C , tRÎ -12¤ where the constant CN can depend on . Hence, æö12¤ \ ()2 + -1¤ 2 \\ ()2 £ sup èøpm A u t A pm u t C . tRÎ \ We tend m ® ¥ to find that any trajectory yt()= ()ut(), u()t lying in the attractor possesses the property \ () = 12¤ \ () £ -¥ <<¥ u t 12¤ A u t C ,.t R0 By virtue of (6.6) we have \ Lu()t £ C ,.-¥ <

Exercise 6.2 In addition to the hypotheses of Theorem 6.3 we assume that ºrÎ º ()l A L 0 and l DA . Prove that the global attractor of ()H, ´ the system St lies in l + 1 l . Conditions of Regularity of Attractor 261

~~§ 7 Conditions of Regularity of Attractor~~

Unfortunately, the structure of the global attractor of problem (0.1) and (0.2) can be described only under additional conditions that guarantee the existence of the Lya- punov function (see Section 1.6). These conditions require that L º 0 and assume the stationarity of the transverse load pt() . For the Berger system (0.3) and (0.4) these hypotheses correspond to r = 0 and px(), t º px() , i.e. to the case of plate oscillations in a motionless stationary medium. Thus, let us assume that the operator L is identically equal to zero and pt()º p in (0.1). Assume that the hypotheses of Theorem 3.1 hold. Then energy equality (3.6) implies that

~~t2 t2 ()()-g()() = - \ ()t 2 t + (), \ ()t 2 t Eyt2 Eyt1 ò u d ò pu d , (7.1)~~

t1 t1 ()==()(), \ () () where yt St yutu t , the function ut is a weak solution to problem = (); () (0.1) and (0.2) with the initial conditions yu0 u1 , and Ey is the energy of the system defined by formula (3.7). Y()= ()- (), = (); Let us prove that the functional y Ey pu0 with yu0 u1 is a Lyapunov function (for definition see Section 1.6) of the dynamical system ()H, Y() H St . Indeed, it is evident that the functional y is continuous on . By vir- ()()= () > tue of (7.1) it is monotonely increasing. If Eyt0 Ey for some t0 0 , then

t0 \ ò u()t 2 dt = 0 . 0 \ ()t =tÎ [], ()= Therefore, u 0 for 0 t0 , i.e. ut u is a stationary solution to prob- = (); lem (0.1) and (0.2). Hence, yu0 is a fixed point of the semigroup St . Therefore, Theorems 1.6.1 and 6.1 give us the following assertion.

~~Theorem 7.1~~

Assume that g > 0 , L º 0 , and p Îss for some > 0 . We also assume that the function Mz() satisfies conditions (3.2) and (5.2). Then the global AH(), attractor of the dynamical system St generated by problem (0.1) and (0.2) has the form

A = M+ ()N , (7.2) N where is the set of fixed points of the semigroup St , i.e. ìü N = (); Î , 2 + æö12¤ 2 = íýu 0 : u 1 A u MAèøu Au p , (7.3) îþ ()N N and M+ is the unstable set emanating from (for definition see Sec- tion 1.6). 262 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h Exercise 7.1 Let p º 0 . Prove that if the hypotheses of Theorem 7.1 hold, a p then any fixed point z of problem (0.1) and (0.2) either equals to ze- t = (); = ()× ; ro, z 00, or has the form zcek 0 , where the constant e ()l2 l + = r cMc is the solution to the equation k k 0 . 4 Exercise 7.2 Assume that p º 0 and Mz()= - G+ z . Then problem (0.1) =Gl(); £ and (0.2) has a unique fixed point z0 00 for 1 . l < Gl£ + Ifn n +1 , then the number of fixed points is equal to 2n 1 = (); = ,,,±¼± and all of them have the form zk wk 0 ,k 01 n , where Gl- w = 0 ,,.w = ± k × e k = 12,,¼ ,n 0 ±k l k k

Exercise 7.3 Show that if the hypotheses of Exercise 7.2 hold, then the () energy Ezk of each fixed point zk has the form Ez()= 0 ,,,Ez()= -Gl1 ()- 2 k = 12,,¼ ,n 0 ±k 4 k l < Gl£ for n n +1 . Exercise 7.4 Assume that the hypotheses of Theorem 7.1 hold. Show that if the set ìü D = íýyu= (); u Î H : Y()y º Ey()- ()pu, £ c (7.4) c îþ0 1 0 is not empty, then it is a closed positively invariant set of the dyna- ()H, mical system St generated by weak solutions to problem (0.1) and (0.2).

Exercise 7.5 Assume that the hypotheses of Theorem 7.1 hold and the set D c defined by equality (7.4) is not empty. Show that the dynamical ()D , A = ()N system c St possesses a compact global attractor c M+ c , N where c is the set of fixed points of St satisfying the condition Y()z £ c .

Exercise 7.6 Show that if the hypotheses of Theorem 7.1 hold, then the A global minimal attractor min (for definition see Section 1.3) of problem (0.1) and (0.2) coincides with the set N of the fixed points (see (7.3)).

Further we prove that if the hypotheses of Theorem 7.1 hold, then the attractor A of problem (0.1) and (0.2) is regular in generic case. As in Section 2.5, the corresponding arguments are based on the results obtained by A. V. Babin and M. I. Vishik (see also Section 1.6). These results prove that in generic case the number of fixed points is finite and all of them are hyperbolic. Conditions of Regularity of Attractor 263

~~Lemma 7.1. Assume that conditions (3.2) and (5.2) are fulfilled. Then the problem~~

2 æö12¤ 2 2 + s L []u º A uMA+ èøu Au = p ,,uDAÎ () (7.5) possesses a solution for any p Îss , where ³ 0 . If B is a bounded set L-1()B = ()2 + s B in s , then its preimage is bounded in 2 + s DA . If -1 2 + s is a compact in s , then L ()B is a compact in DA() , i.e. the mapping L is proper.

Proof. We follow the line of arguments given in the proof of Lemma 2.5.3. Let us consider the continuous functional ìü ()= 1 (), + M æö12¤ 2 - (), Wu íýAu Au èøA u pu (7.6) 2 îþ z = () M ()= ()xx () on 1 DA , where z ò M d is a primitive of the function Mz . Equation (3.2) implies that 0 æö Wu()³³1 Au 2 - aA12¤ u 2 - b - A-1pAu× 2 èø - 1 æö æö1 2 ³ 1 - a Au 2 - b - 1 - a A-1p . (7.7) 4 èøl 2 èøl 1 1 Thus, the functional Wu() is bounded below. Let us consider it on the subspace {},,¼ pm 1 , where pm is the orthoprojector onto Lin e1 em as before. Since ()® +¥ ® ¥ Wu as Au , there exists a minimum point um on the subspace pm 1 . This minimum point evidently satisfies the equation

2 + æö12¤ 2 = A um MAèøum Aum pm p . (7.8) Equation (7.7) gives us that ìü 2 () Î -1 2 Au £ c ++c inf íýWu: upm 1 c A p m 1 2 îþ3 with the constants being independent of m . Therefore, it follows from (7.8) 2 £ £ that A um CR , provided pR . This estimate enables us to pass to the limit in (7.8) and to prove that if s = 0 , then equation (7.5) is solvable for any p Î s . Equation (7.5) implies that 2 + s £ £ A um CR forp s R , i.e. L-1()B is bounded in DA()2 + s if B is bounded. In order to prove that the mapping L is proper we should reason as in the proof of Lemma 2.5.3. We give the reader an opportunity to follow these reasonings individually, as an exercise. Lemma 7.1 is proved. 264 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h Lemma 7.2 a Î L ¢ [] p Let u 1 . Then the operator u defined by the formula t æö æö e L ¢ []u wA= 2 w ++2 M ¢ A12¤ u 2 ()Au, w Au M A12¤ u 2 Aw (7.9) r èø èø 4 with the domain D()L ¢[]u = DA()2 is selfadjoint and dim Ker L ¢[]u < < ¥ .

Proof. It is clear that L ¢[]u is a symmetric operator on DA()2 . Moreover, it is easy to verify that L ¢[]u wA- 2 w £ Cu()× Aw ,,(7.10)wDAÎ ()2 i.e. L ¢[]W is a relatively compact perturbation of the operator A2 . Therefore, L ¢[]u is selfadjoint. It is further evident that

~~Ker L ¢[]= Ker + -2 æöL ¢[]- 2 u IAèøu A .~~

~~However, due to (7.10) the operator A-2 ()L ¢[]u - A2 is compact. Therefore, dim Ker L ¢[]u < ¥ . Lemma 7.2 is proved.~~

Î L ¢[] Exercise 7.7 Prove that for any u 1 the operator u is bounded below and has a discrete spectrum, i.e. there exists an orthonormal {} H basis fk in such that L ¢[] = m =m,,¼ ££m ¼ m= ¥ u fk k fk ,,k 12 1 2 lim n . n ® ¥

Exercise 7.8 Assume that uc= e , where c is a constant and e is an 0 k0 0 k0 {} element of the basis ek of eigenfunctions of the operator A . Show L ¢[] = m = ,,¼ that u ek k ek for all k 12 , where m = l2 12+ d c2 M ¢()c2 l + Mc()l2 l . k k kk0 0 0 k0 0 k0 k Here d = 1 for kk=d and = 0 for kk¹ . kk0 0 kk0 0 As in Section 2.5, Lemmata 7.1 and 7.2 enable us to use the Sard-Smale theorem (see, e.g., the book by A. V. Babin and M. I. Vishik [10]) and to state that the set ì ü -1 -1 Rs = íh Î s : $ []L ¢[]u for all u Î L []h ý î þ

~~of regular values of the operator L is an open everywhere dense set in s for s ³ 0 .~~

~~Exercise 7.9 Show that the set of solutions to equation (7.5) is finite for p ÎRs (Hint: see the proof of Lemma 2.5.5). Conditions of Regularity of Attractor 265~~

Let us consider the linearization of problem (0.1) and (0.2) on a solution uDAÎ ()2 to problem (7.5): \\ \ w ++g w L ¢[]u w = 0 , \ (7.11) w = w , w = w . t = 0 0 t = 0 1 Here L ¢[]u is given by formula (7.9).

~~Exercise 7.10 Prove that problem (7.11) has a unique weak solution on any [], Î Î ()Î segment 0 T if w0 1 , w1 0 , and the function Mz 2 Î C ()R+ possesses property (3.2).~~

Thus, problem (7.11) defines a strongly continuous linear evolutionary semigroup [] H= ´ Tt u in the space 1 0 by the formula [](); = ()(); \ () Tt u w0 w1 wt w t ,(7.12) where wt() is a weak solution to problem (7.11).

{} Exercise 7.11 Let fk be the orthonormal basis of eigenelements of the L ¢[] m operator u and let k be the corresponding eigenvalues. Then each subspace H = {}(); , (); Ì H k Lin fk 0 0 fk [] is invariant with respect to Tt u . The eigenvalues of the restriction [] H of the operator Tt u onto the subspace k have the form ìüg g2 -æö ± - m exp íýèøk t . îþ2 4

Lemma 7.3 2 Let L º 0 . Assume that Mz()Î C ()R+ possesses property (3.2). Then the evolutionary operator St of problem (0.1) and (0.2) is Frechét dif- = (); ¢ []= [] ferentiable at each fixed point yu0 . Moreover, St u Tt u , [] where Tt u is defined by equality (7.12).

Proof. Let ()= []+ - - [] zt St yh y Tt u h , where hh= (); h Î H , yu= (); 0 , and u is a solution to equation (7.5). 0 1 \ It is clear that zt()= ()vt(); v()t , where vt()º ut()- u - wt() is a weak solution to problem \\ ì v ++g vA2 v = Fut()(),,uwt() , í \ (7.13) v = 0, v = 0 . î t = 0 t = 0 266 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h Here a æöæö p Fut()(),,uwt() = MA12¤ u 2 Au- M A12¤ ut()2 Au()+ t t èøèø e r æö æö + MA12¤ u 2 Aw() t + 2 M ¢ A12¤ u ()Au, w() t Au , 4 èø èø where ut() is a weak solution to problem (0.1) and (0.2) with the initial condi- = + º ()+ ; () tions y0 yh uh0 h1 and wt is a solution to problem (7.11) with = = w0 h0 and w1 h1 . It is evident that ()(),,() = - æö12¤ 2 + ()- ¢æö12¤ 2 () Fut uwt MAèøu Av F1 t M èøA u F2 t ,(7.14) where ì ()= - æö12¤ ()2 --æö12¤ 2 F1 t íMAèøut MAèøu î ü - ¢æö12¤ 2 æö12¤ 2 - 12¤ 2 () M èøA u èøA ut A u ý Au t , þ

~~()= æö12¤ ()2 - 12¤ 2 ()- (), () F2 t èøA ut A u Au t 2 Au w t Au . () It is also evident that the value F1 t can be estimated in the following way~~

ìü 2 F ()t £ c × max íýM ¢¢ ()z : z Î []0, c A12¤ ut()2 - A12¤ u 2 1 1 îþ2 Î [], £ for t 0 T and for h H R , where the constants c1 and c2 depend on T , Ru, and . This implies that () £ (),, ()()- 2 F1 t CT R u Aut u .(7.15) () Let us rewrite the value F2 t in the form ()= ()()+ , ()()- × ()()- + F2 t ut u Aut u Aut u

~~++A12¤ ()ut()- u 2 Au 2 ()uAvt, () Au . Consequently, the estimate () £ (),, ()()- 2 + () () F2 t C1 TRu Aut u C2 u Av t (7.16)~~

holds for t Î []0, T and for h H £ R . Therefore, equations (7.14)–(7.16) give us that ()(),, () £ () + ()()- 2 Fut uwt C1 Av t C2 Aut u [], = () = (),, on any segment 0 T . Here C1 C1 u and C2 C2 TRu . We use continuity property (3.20) of a solution to problem (0.1) and (0.2) with respect to the initial conditions to obtain that Conditions of Regularity of Attractor 267

~~()()- £ Î [], £ Aut u CTR, h H ,,.t 0 T h H R Therefore, ()(),, () £ () + 2 Fut uwt C1 Av t C2 h H~~

for t Î []0, T and for h H £ R . Hence, the energy equality for the solutions to problem (7.13) gives us that æö\ æö\ d Av() t 2 + v()t 2 £ aAvt()2 + v()t 2 + Ch4 . dt èøèø Therefore, Gronwall’s lemma implies that \ Av() t 2 + v()t 2 £ Ch4 ,.t Î []0, T This equation can be rewritten in the form []+ - - [] £ 2 St yh y Tt u h H Ch . Thus, Lemma 7.3 is proved.

Exercise 7.12 Use the arguments given in the proof of Lemma 7.3 to verify 2 that under condition (3.2) for Mz()Î C ()R+ the evolutionary H 1 operator St of problem (0.1) and (0.2) in belongs to the class C and ¢ []- ¢ [] £ - St y1 St y2 LH(), H Cy1 y2 H > Î H for any t 0 and yj . Exercise 7.13 Use the results of Exercises 7.7 and 7.11 to prove that for a regular value p of the mapping L []u the spectrum of the opera- [] tor Tt u does not intersect the unit circumference while the eigensubspace E+ which corresponds to the spectrum outside the unit disk does not depend on t and is finite-dimensional.

~~The results presented above enable us to prove the following assertion (see Chap- ter V of the book by A. V. Babin and M. I. Vishik [10]).~~

Theorem 7.2 Assume that the hypotheses of Theorem 7.1 hold. Then there exists an R ()H, open dense set s in s such that the dynamical system St possesses a regular global attractor A for every p Î Rs , i. e. N A = () ÈM+ zj , j = 1 () where M+ zj is the unstable manifold of the evolutionary operator St ema- () nating from the fixed point zj . Moreover, each set M+ zj is a finite-dimensional surface of the class C1 . 268 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C ()º h In the case of a zero transverse load p 0 Theorem 7.2 is not applicable in gene- a ral. However, this case can be studied by using the structure of the problem. For exam- p t ple, we can guarantee finiteness of the set of fixed points if we assume (see Exer- e cise 7.1) that the equation Mc()l2 l + = 0 , first, is solvable with respect to c r k k l only for a finite number of the eigenvalues k and, second, possesses not more than 4 a finite number of solutions for every k . The solutions to equation (7.5) are either 2 u º 0 , or uc= ek , where c and k satisfy Mc()ll + = 0 . The eigen- 0 0 0 0 0 k0 k0 values of the operator L ¢[]u have the form m = l2 + ()l º k k M 0 k if u 0 and

m = l l - l + 2 d l c2 M ¢()c2 l ifuc= e . k k k k0 kk0 k0 0 0 k0 0 k0 Therefore, the result of Exercise 7.11 implies that the fixed points are hyperbolic m if all the numbers k are nonzero, i.e. if M ()0 ¹ - l,;,;k =l12,,¼ ¹ l kk¹ Mc()2 l ¹ 0 k k k0 0 0 k0 for all c and k such that Mc()l2 l + = 0 . In particular, if Mz()= - G+ z , 0 0 0 k0 k0 then for any real G there exists a finite number of fixed points (see Exercise 7.2) Gl¹ and all of them are hyperbolic, provided that k for all k and the eigenvalues l l < G j satisfying the condition j are simple. Moreover, we can prove that for l <

~~§ 8 On Singular Limit in the Problem of Oscillations of a Plate~~

~~In this section we consider problem (0.1) and (0.2) in the following form:~~

m \\ ++g \ 2 +æö12¤ 2 + = , > u u A uMAèøu Au Lu p t 0, (8.1) \ u = u , u = u .(8.2) t = 0 0 t = 0 1 Equation (8.1) differs from equation (0.1) in that the parameter m > 0 is introduced. It stands for the mass density of the plate material. The introduction of a new time t ¢ = t ¤ m transforms equation (8.1) into (0.1) with the medium resistance parameter g¢= g¤ m instead of g . Therefore, all the above results mentioned above remain true for problem (8.1) and (8.2) as well. On Singular Limit in the Problem of Oscillations of a Plate 269

The main question discussed in this section is the asymptotic behaviour of the solution to problem (8.1) and (8.2) for the case when the inertial forces are small with respect to the medium resistance forces mg . Formally, this assumption leads to a quasistatic statement of problem (8.1) and (8.2):

\ 2 æö12¤ 2 g u ++A uMAèøu Au + Lu = p, t > 0, (8.3) u = u . (8.4) t = 0 0 Here we prove that the global attractor of problem (8.1) and (8.2) is close to the global attractor of the dynamical system generated by equations (8.3) and (8.4) in some sense. Without loss of generality we further assume that g = 1 . We also note that problem (8.3) and (8.4) belongs to the class of equations considered in Chapter 2.

Exercise 8.1 Assume that conditions (3.2) and (3.3) are fulfilled and Î = p 0 H . Show that problem (8.3) and (8.4) has a unique mild = () [], (in 1 DA ) solution on any segment 0 T , i.e. there exists ()Î (), ; a unique function ut C 0 T 1 such that ()= -A2t - ut e u0

~~t ìü - -A2 ()t - t æö12¤ ()t 2 ()t + ()t - t e íýMAèøu Au Lu p d . ò îþ 0 (Hint: see Theorem 2.2.4 and Exercise 2.2.10).~~

~~Let us consider the Galerkin approximations of problem (8.3) and (8.4):~~

\ ()++2 () æö12¤ ()2 +() = um t A um t MAèøum t Aum pm Lum t pm p , (8.5) ()= um 0 pm u0 , (8.6) where pm is the orthoprojector onto the first mA eigenvectors of the operator and ()Î {},,¼ um t Lin e1 em . Exercise 8.2 Assume that conditions (3.2) and (3.3) are fulfilled and Î = p 0 H . Then problem (8.5) and (8.6) is solvable on any segment []0, T and ()()- () ® ® ¥ maxAut um t 0 ,.(8.7)m []0, T

Theorem 8.1 Let pHÎ and assume that conditions (3.2), (5.2), and (5.3) are ful- () , filled. Then the dynamical system 1 St generated by weak solutions to problem (8.3) and (8.4) possesses a compact connected global attractor A . £ b < This attractor is a bounded set in 1 + b for 0 1 and has a finite fractal dimensiondimension.. 270 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h Proof. a First we prove that the system () , S is dissipative. To do that we consider p 1 t () t the Galerkin approximations (8.5) and (8.6). We multiply (8.5) by um t scalarwise e r and find that 1 æö 4 d u ()t 2 ++Au ()t 2 MA12¤ u ()t 2 A12¤ u ()t 2 + 2 dt m m èøm m + ()(), () = (), () Lum t um t pum t . Using equation (5.2) we obtain that

2 æö 1 d u ()t 2 ++Au ()t a M A12¤ u ()t 2 £ 2 dt m m 1 èøm £ - 12¤ ()22+ a - ()(), () + (), () a3 a2 A um t Lum t um t pum t . We use equation (5.3) and reason in the same way as in the proof of Theorem 5.1 to find that æöæö + a 1 d u 2 ++b Au 2 + M A12¤ u 2 b A12¤ u 22 £ b (8.8) 2 dt m 0 èøm èøm 1 m 2 = ,, \ () with some positive constants bj , j 012 . Multiplying equation (8.5) by um t we obtain that \ \ \ 1 d P()u ()t ++u ()t 2 ()Lu ()t , u ()t = ()pu, ()t , 2 dt m m m m m where

~~P()= 2 + M æö12¤ 2 u Au èøA u . It follows that \ d P()u ()t + u ()t 2 £ 2 p 2 + CAq u ()t 2 . (8.9) dt m m m If we summarize (8.8) and (8.9), then it is easy to find that~~

~~ìüìü d íýu 2 + P()u + b íýu 2 + P()u £ C . dt îþm m 0 îþm m This implies that~~

~~ìü-b t u ()t 2 + P()u ()t £ íýp u 2 + P()p u e 0 + C . m m îþm 0 m 0 We use (8.7) to pass to the limit as m ® ¥ and to obtain that~~

ìü-b t ut()2 + P()ut() £ íýu 2 + P()u e 0 + C . îþ0 0 () , This implies the dissipativity of the dynamical system 1 St generated by problem (8.3) and (8.4). In order to complete the proof of the theorem we use Theorem 2.4.1. On Singular Limit in the Problem of Oscillations of a Plate 271

~~() , We note that the dissipativity also implies that the dynamical system 1 St possesses a fractal exponential attractor (see Theorem 2.4.2).~~

Exercise 8.3 Assume that the hypotheses of Theorem 8.1 hold and L º 0 . Show that for generic pHÎ the attractor of the dynamical system () , 1 St generated by equations (8.3) and (8.4) is regular (see the definition in the statement of Theorem 7.2). Hint: see Section 2.5.

2 We assume that Mz()Î C ()R+ and conditions (3.2), (5.2), (5.3), and (5.10) are ()H , m fulfilled. Let us consider the dynamical system 1 St generated by problem H = ´ º ()2 ´ () (8.1) and (8.2) in the space 1 2 1 DA DA . Lemma 5.2 and Exer- ()H , m A cise 5.5 imply that 1 St possesses a compact global attractor m for any m > 0 . The main result of this section is the following assertion on the closeness of attractors of problem (8.1) and (8.2) and problem (8.3) and (8.4) for small m > 0 .

Theorem 8.2 2 Assume that Mz()Î C ()R+ and conditions (3.2), (5.2), (5.3), and (5.10) concerning Mz(), Lp, and are fulfilled... Then the equation * limsup {}distH ()y, A : y ÎAm = 0 (8.10) m ® 0 A ()H , m is valid, where m is a global attractor of the dynamical system 1 St generated by problem (8.1) and (8.2), ìü A* = (); ÎA = - 2 - æö12¤ 2 - + íýz0 z1 : z0 , z1 A z0 MAèøz0 Az0 Lz0 p . îþ A Here is a global attractor of problem (8.3) and (8.4) in 1 and distH ()yA, is the distance between the element y and the set A in the H=g´ = space 1 0 . We remind that 1 in equations (8.1) and (8.3).

~~The proof of the theorem is based on the following lemmata.~~

Lemma 8.1 ()H , m H The dynamical system 1 St is uniformly dissipative in with m Îm( ,]m >m> respect to 0 0 for some 0 0 , i.e. there exists 0 0 and > Ì H H R 0 such that for any set B 1 which is bounded in we have m ìü S ByuÌ íý= (); u : m u 2 + Au 2 £ R2 (8.11) t îþ0 1 1 0 ³m(), m Î ( ,]m for all ttB ,0 0 . 272 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h Proof. a We use the arguments from the proof of Theorem 5.1 slightly modifying p t them. Let e ()= ()+ nB() = (); r Vy Ey y ,,yu0 u1 4 where æöæö Ey()= 1 m u 2 ++Au 2 M A12¤ u 2 2 èø1 0 èø0 and

B()y = m()u , u + 1 u 2 . 0 1 2 0 As in the proof of Theorem 5.1 it is easy to find that the inequalities e d Eyt()() £ - 1 u 2 ++ Au 2 C A12¤ u 2 +C (8.12) dt 2 2 e 1 and \ æö d B()myt() £ u 2 - 1 Au 2 - a M A12¤ u 2 - dt 2 1 èø - 12¤ 22+ a ++2 a2 A u C2 u C3 (8.13) ()==e()(); \ () m > are valid for yt ut u t St y0 . Here 0 is an arbitrary number, m the constants aj and cj do not depend on . Moreover, it is also evident that æöæö Vy()£ 1 b m u 2 ++Au 2 M A12¤ u 2 + b (8.14) 2 0 èø1 0 èø0 1 m Îm( ,]m b b mÎ for 0 0 and for any 0 . Here 0 and 1 do not depend on În( ,]m £ 0 0 and 1 . Equations (8.12)–(8.14) lead us to the inequality d Vyt()d() + Vyt()() £ --1 ()1 - ()mdb + n u\ 2 dt 2 0 æödb æö -n1 ()ndb-e- Au 2 - a - 0 M A12¤ u 2 + D , 2 0 èø1 2 èø D m Îm( ,]m where the constant does not depend on 0 0 . If we choose 0 small d >n>mÎ ( ,]m enough, then we can take 0 and 0 independent of 0 0 and such that

d Vyt()d() + Vyt()() £ D ,(8.15) dt 1 D >mÎ ( ,]m where 1 0 does not depend on 0 0 . Moreover, we can assume m (due to the choice of 0 ) that ()b() ³ æöm \ ()2 + ()2 - Vyt 3 èøu t Au t C ,(8.16) b m Î ( ,]m where 3 and C do not depend on 0 0 . Using equations (8.14)–(8.16) we obtain the assertion of Lemma 8.1. On Singular Limit in the Problem of Oscillations of a Plate 273

Lemma 8.2 Let ut() be a solution to problem (8.1) and (8.2) such that Au() t £ R for all t ³ 0 . Then the estimate t 1 \ 2 -b ()- t æö\ 2 2 -b u()t e t dt £ m u()0 + Au()0 e t + CR(), b 2 ò èø 0 is valid for t ³b0 , where is a positive constant such that bm£ 12¤ .

~~Proof. It is evident that the estimate~~

æö12¤ ()2 ()+ ()- £ MAèøut Au t Lu t p CR holds, provided Au() t £ R . Therefore, equaiton (8.1) easily implies the estimate æö\ \ 1 d m u()t 2 + Au() t 2 + 1 u()t 2 £ C 2 dt èø2 R for the solution ut() . We multiply this inequality by 2 exp()b t . Then by virtue of the fact that Au() t £ R we have æö\ \ d eb t m u()t 2 + Au() t 2 + 1 u()t 2 e b t £ C eb t dt èø2 R, b for mb £ 12¤ . We integrate this equation from 0 to t to obtain the assertion of the lemma.

Lemma 8.3 Let ut() be a solution to problem (8.1) and (8.2) with the initial condi- (); Î H = ´ () £ ³ tions u0 u1 1 2 1 and such that Au t R for t 0 . Then the estimate -b t m \ ()2 + ()2 £ æö+ \ ()2 + ()2 0 + w t Aw t C1 èø1 w 0 Aw 0 e C2 (8.17) ()=m\ () Îm( ,]m is valid for the function wt u t . Here 0 0 , 0 is small b > enough, 0 0 , and the numbers C1 and C2 do not depend on m Î ( ,]m 0 0 .

Proof. Let us consider the function æö\ æö\ Wt()= 1 m w()t 2 + Aw() t 2 + nm()w, w + 1 w 2 2 èøèø2 for n > 0 . It is clear that 274 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h \ 1 m ()1 - nm w 2 + 1 Aw 2 ££Wt() a 2 2 p t £ 1 m()+ nm \ 2 + æö1 + nl-2 2 e 1 w èø1 Aw . (8.18) r 2 2 4 Since the function wt() is a weak solution to the equation obtained by the differentiation of (8.1) with respect to t (cf. (4.15)):

~~m \\ ()++\ () 2 () +æö12¤ ()2 () +()= () w t w t A wt MAèøut Aw t Lw t Ft , where~~

æö12¤ 2 Ft()= -2 M ¢èøA ut() ()Au() t , wt() Au() t , then we have that \ \ \ \ d Wt() = - ()1 - nm w 2 - MA()12¤ ut()2 ()Aw, w - ()Lw, w + ()-Fw, dt ìü - n íýA2 w 2 ++MA()12¤ ut()2 A12¤ w 2 ()Lw, w - ()Fw, . îþ It follows that

~~æö\ ()1 ()2 d Wt() £ - 1 - nm w 2 -n1 ()n - C Aw 2 + C w 2 . dt èø2 2 R R n =m()1 + We take CR 2 and choose 0 small enough to obtain with the help of (8.18) that~~

d Wt()+ b Wt() £ Cwt()2 ,,t ³m0 Î (0,]m , dt 0 0 b > m where the constants 0 0 and C do not depend on . Consequently, t -b t \ -b ()t - t Wt() £ W()0 e 0 + Cuò ()t 2 e 0 dt . 0 Therefore, estimate (8.17) follows from equation (8.18) and Lemma 8.2. Thus, Lemma 8.3 is proved.

Lemma 8.3 and equations (8.11) imply the existence of a constant R1 such that for H = (), m any bounded set B in 1 there exists t0 t0 B such that m \\ ()2 + \ ()2 £ 2 m Î (), m u t Au t R1 ,,0 0 (8.19) where ut() is a solution to problem (8.1) and (8.2) with the initial conditions from BA. However, due to (8.1) equations (8.11) and (8.19) imply that 2ut() £ C for ³ (), m > tt0 B . Thus, there exists R2 0 such that On Singular Limit in the Problem of Oscillations of a Plate 275

m \\ ()2 ++\ ()2 2 ()2 £ 2 ³ (), m u t Au t A ut R2 ,,tt0 B (8.20) where ut() is a solution to system (8.1) and (8.2) with the initial conditions from the H m Îm(), m bounded set B in 1 , R2 does not depend on 0 0 , and 0 is small enough. Equation (8.20) and the invariance property of the attractor Am imply the estimate m \\ ()2 ++\ ()2 2 ()2 £ 2 um t Aum t A um t R2 (8.21) m = ()(); \ () A Î -¥¥ for any trajectory St y0 um t um t lying in m for all t . = (); Let us prove (8.10). It is evident that there exists an element ym u0 m u1 m from Am such that ()º (), A* = {}(), A* ÎA dym distH ym sup distH y : y m . \ Let ym()t = ()um()t ; um()t be a trajectory of system (8.1) and (8.2) lying in the attractor Am and such that ym()0 = ym . Equation (8.21) implies that there exist \ a subsequence {}y ()t and an element yt()= ()ut(); u()t Î L¥ ()-¥¥, ; H mn 1 such that for any segment []ab, the sequence y ()t converges to yt() in the mn ¥ ()m, ; H ® * -weak topology of the space L ab 1 as n 0 . Equation (8.21) gives us that the subsequence {}Au ()t is uniformly continuous and uniformly bounded mn inH . Therefore (cf. Exercise 1.14),

~~limmax Au()m ()t - ut() = 0 (8.22) m ® Î [], n n 0 tab \\ for any ab~~

~~dy()m £ ym - y ® 0 ,,m ® 0 n n 0 H n where~~

~~= æö(); - ()- æö12¤ ()2 - + Î A* y0 èøu 0 Au 0 MAèøu 0 Au0 Lu0 p . Thus, Theorem 8.2 is proved. 276 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow~~

C h § 9 On Inertial and Approximate a p Inertial Manifolds t e r The considerations of this section are based on the results presented in Sections 3.7, 4 3.8, and 3.9. For the sake of simplicity we further assume that pt()º g Î H .

~~Theorem 9.1 Assume that conditions (3.2), (5.2), and (5.3) are fulfilled. We also assume that eigenvalues of the operator A possess the properties~~

l r infN >l 0 and =rc k ()1+ o()1 ,,, > 0 ,,,k ® ¥ , (9.1) l Nk()+1 0 N N +1 {}¥()®g> > for some sequence Nk . Then there exist numbers 0 0 and k0 0 such that the conditions gg>l2 - l2 ³ l 0 and Nk()+ 1 Nk() k0 Nk()+1 (9.2) ()H, imply that the dynamical system St generated by problem (0.1) and (0.2) possesses a local inertial manifold, i.e. there exists a finite-dimen- MH= ´ sional manifold in 1 0 of the form M = {}yw= + B()w : wPÎ H , B()w Î ()1 - P H , (9.3) where B(). is a Lipschitzian mapping from PH into ()1- P H and P is a finite-dimensional projector in H . This manifold possesses the properties: H ³ () 1) for any bounded set B in and for tt0 B {}(), M Î B £ {}-b()- () sup distSt y : y C exp tt0 B ; (9.4) > ÎM £ 2) there exists R 0 such that the conditions y and St y H R for Î [], ÎM Î [], t 0 t0 imply that St y for t 0 t0 ; ()H, M 3) if the global attractor of the system St exists, then the set contains it (see Theorem 6.1)..

~~Proof. Conditions (3.2), (5.2), and (5.3) imply (see Theorem 5.1) that the dynamical ()H, > system St is dissipative, i.e. there exists R 0 such that £ Î ³ () St y H R ,,yB tt0 B (9.5)~~

Î H ()H, for any bounded set B . This enables us to use the dynamical system St generated by an equation of the type \\ ++g \ 2 = () ì u u A u BR u , í \ (9.6) î u = u , u = u , t = 0 0 t = 0 1 to describe the asymptotic behaviour of solutions to problem (0.1) and (0.2). Here

~~- ìü ()= cæö()1 × - æö12¤ 2 - BR u èø2R Au íýgMAèøu Au Lu îþ On Inertial and Approximate Inertial Manifolds 277~~

~~and c()s is an infinitely differentiable function on R+ possessing the properties 0 ££c()s 1 ;; c¢()s £ 2 c()s = 1 ,;0 ££s 1 c()s = 0 ,.s ³ 2~~

~~It is easy to find that there exists a constant CR such that ()£ BR u CR and ()- ()£ ()- BR u1 BR u2 CR Au1 u2 .~~

~~()H, Therefore, we can apply Theorem 3.7.2 to the dynamical system St generated by equation (9.6). This theorem guarantees the existence of an inertial manifold of~~

()H, the system St if the hypotheses of Theorem 9.1 hold. However, inside the dissipativity ball {}y : y H £ R problem (9.6) coincides with problem (0.1) and (0.2). This easily implies the assertion of Theorem 9.1.

Exercise 9.1 Show that the hypotheses of Theorem 9.1 hold for the problem on oscillations of an infinite panel in a supersonic flow of gas: ì æöp ï ¶2 +++g¶ ¶4 ç÷G¶- (), 2 ¶2 +r¶ = () Î(), p , > ï t u t u x u ç÷ò xux t dx x u xu gx, x 0 t 0, í èø ï 0 ï ==¶2 = ¶ = u = , = p x u = , = p 0 , u = u0 x , t u u1 x . î x 0 x x 0 x t 0 t = 0 Here Gr and are real parameters and gx()Î L2()0, p .

It is evident that the most essential assumption of Theorem 9.1 that restricts its application is condition (9.2). In this connection the following assertion concerning the case when problem (0.1) and (0.2) possesses a regular attractor is of some interest.

Theorem 9.2 Assume that in equation (0.1) we have L º 0 and pt()ºÎg ÎLin {}e ,,¼ e for some N . We also assume that conditions (3.2) and 1 N0 0 ³ ³ (5.2) are fulfilled. Then there exists N1 N0 such that for all NN1 the subspace H = {}(); , (); = ,,¼ , N Lin ek 0 0 ek : k 12 N (9.7) is an invariant and exponentially attracting set of the dynamical system ()H, St generated by problem (0.1) and (0.2): g - ()tt- ()B (), H £ ()- 4 0 Î dist St y N CB 1 PN y H e , yB (9.8) ³ () H for tt0 B and for any bounded set B in . Here PN is the orthoprojector H onto N . 278 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow

C h Proof. a Since p gg= for NN³ , where p is the orthoprojector onto the span of p N 0 N {},,¼ H t e1 eN , the uniqueness theorem implies the invariance of N . Let us prove e 0 ()(), \ () r attraction property (9.8). It is sufficient to consider a trajectory ut u t lying in the ball of dissipativity {}y : y £ R . Evidently the function vt()= ()1 - p ut() 4 H N satisfies the equation ì \\ ++g \ 2 +æö12¤ ()2 = ï v v A vMAèøut Av 0 , í (9.9) \ ï v = ()1 - p u , v = ()1 - p u . î t = 0 N 0 t = 0 N 1 It is also clear that the conditions æö æö MA12¤ ut()2 £ b ()R and d MA12¤ ut()2 £ b ()R èø0 dt èø1 hold in the ball of dissipativity. This fact enables us to use Theorem 2.2 with bt()= = MA()12¤ ut()2 . In particular, equation (2.23) guarantees the existence of a num- ³g() () ber N1 N0 which depends on , b0 R , and b1 R and such that g - t () = ()- () £ ()- () 4 > yt H 1 PN yt H 31PN y 0 e ,,t 0 ³ H ()= ()(), \() for all NN1 , where PN is the orthoprojector onto N and yt vt v t . This implies estimate (9.8). Theorem 9.2 is proved.

~~Exercise 9.2 Assume that the hypotheses of Theorem 9.2 hold. Show that~~

for any semitrajectory St y there exists an induced trajectory in H ÎH N , i.e. there exists y N such that g - ()tt- - £ 4 0 St ySt y H CR e ³ = () for tt0 and for some t0 t0 y H . Exercise 9.3 Write down an inertial form of problem (0.1) and (0.2) in the H subspace N , provided the hypotheses of Theorem 9.2 hold. Prove that the inertial form coincides with the Galerkin approximation of the order N of problem (0.1) and (0.2).

Exercise 9.4 Show that if the hypotheses of Theorem 9.2 hold, then the global attractor of problem (0.1) and (0.2) coincides with the global attractor of its Galerkin approximation of a sufficiently large order.

Let us now turn to the question on the construction of approximate inertial manifolds for problem (0.1) and (0.2). In this case we can use the results of Section 3.8 and the theorems on the regularity proved in Sections 4 and 5. On Inertial and Approximate Inertial Manifolds 279

m +1 Exercise 9.5 Assume that Mz()Î C ()R+ , m ³ 1 and ()= - æö12¤ 2 - Bu pMAèøu Au Lu, Î Î D ()= (). where pH and u A 1 . Show that the mapping B has the Frechét derivatives B()k up to the order l inclusive. Moreo- ver, the estimates k áñB()k []u ; w ,,¼ w £ C Aw (9.10) 1 k R Õ j j = 1 and áñ()k []- ()k []* ; ,,¼ £ B u B u w1 wk

k £ ()- * CR Au u Õ Awj (9.11) j = 1 are valid, where k = 01,,¼ ,m , Au£ R , Au* £ R , and Î D () áñ()k []; ,,¼ wj A . Here B u w1 wk is the value of the ,,¼ Frechét derivative on the elements w1 wk .

We consider equations (9.10) and (9.11) as well as Theorem 5.3 which guarantees nonemptiness of the classes LmR, corresponding to the problem considered when R > 0 is large enough. They enable us to apply the results of Section 3.8. {},,¼ Let Pe be the orthoprojector onto the span of elements 1 eN in H and = - {}(), \ ¥ {}(), \ ¥ let Q 1 P . We define the sequences hn pp n = 0 and ln pp n = 0 of mappings from PH´ PH into QH by the formulae (), \ ==(), \ h0 pp l0 pp 0 , (9.12) 2 (), \ = - (), \ -g()+ - - A hk pp a0 Mk -1 ppAhk -1 QL p hk -1 ln ()k

~~-áñdd ; \ + áñ; g \ ++2 - (), \ +()+ plk -1 p p\ lk -1 p A pb0 Mk -1 ppAp PL p hk -1 , (9.13) ()d, \ = áñ-; \ lk pp p hk -1 p~~

- áñd ; g \ ++2 - (), \ +()+ p\ hk -1 p A pb0 Mk -1 ppAp PL p hk -1 . (9.14) (), \ = æö12¤ 2 + 12¤ (), \ 2 d d Here Mk pp MAèøp A hk pp , p and p\ are the Frechét de- = = rivatives with respect to the corresponding variables, a0 Qg , b0 Pg , where gptº () is a stationary transverse load in (0.1), k = 12,,¼ ,m , the numbers nk are chosen to fulfil the inequality k -n1 ££()k k .

~~(), \ (), \ Exercise 9.6 Evaluate the functions h1 pp and l1 pp . 280 The Problem on Nonlinear Oscillations of a Plate in a Supersonic Gas Flow~~

C h Theorem 3.8.2 implies the following assertion. a p t Theorem 9.3 e ()º Î ()Î m + 1() ³ r Assume that pt g H , Mz C R+ , m 2 , and conditions (3.2), (5.2), and (5.3) are fulfilled. Then for all k = 1,,¼ m the collection of 4 (), mappings hn ln given by equalities (9.12)–(9.14) possesses the properties = (), r (), r = , 1) there exist constants Mj Mj n and Lj n , j 12 such that 2 (), \ £ (), \ £ A hn p0 p0 M1 ,,,Aln p0 p0 M2 , 2()(), \ - (), \ £ ()2 ()- + ()\ - \ A hn p1 p1 hn p2 p2 L1 A p1 p2 Ap1 p2 , ()(), \ - (), \ £ ()2()- + ()\ - \ Aln p1 p1 ln p2 p2 L2 A p1 p2 Ap1 p2 , \ for all pj and pj from PH and such that 2 2 +r\ 2 £ =r,, > A pj Apj , j 012, 0 ; 2) for any solution ut() to problem (0.1) and (0.2) which satisfies compatibility conditions (4.3) with lm= the estimate ìü12¤ 2 2 2 -n íýA ()ut()- u ()t + Aut()()- u ()t £ C l + îþn n n N 1 is valid for nm£ -1 and for t large enough. Here ()= ()+ ()(), \ () un t pt hn pt p t , ()= \ ()+ ()(), \ () un t p t ln pt p t , l N is the N -th eigenvalue of the operator A and the constant Cn depends on the radius of dissipativity.

In particular, Theorem 9.3 means that the manifold M = {}()+ (), \ ; \ + (), \ , \ Î n phn pp p ln pp : pp PH ()H, attracts sufficiently smooth trajectories of the dynamical system St generated l-n M by problem (0.1) and (0.2) into a small vicinity (of the order Cn N +1 ) of n .

Exercise 9.7 Assume that the hypotheses of Theorem 6.3 hold (this theorem guarantees the existence of the global attractor A consisting of smooth trajectories of problem (0.1) and (0.2)). Prove that {}(), M ÎA £ l-n sup disty n : y Cn N +1 for all nm£ -1 (the number m is defined by the condition m + 1 Mz()Î C ()R+ ). Exercise 9.8 Prove the analogue of Theorem 3.9.1 on properties of the nonlinear Galerkin method for problem (0.1) and (0.2). References

1. BERGER M. A new approach to the large deflection of plate // J. Appl. Mech. –1955. –Vol. 22. –P. 465–472. 2. ZH.N.DMITRIEVA. On the theory of nonlinear oscillations of thin rectangular plate // J. of Soviet Mathematics. –1978. –Vol. 22. –P. 48–51. 3. BOLOTIN V. V. Nonconservative problems of elastic stability. –Oxford: Pergamon Press, 1963. 4. DOWELL E. H. Flutter of a buckled plate as an example of chaotic motion of deterministic autonomous system // J. Sound Vibr. –1982. –Vol. 85. –P. 333–344. 5. YOSIDA K. Functional analysis. –Berlin: Springer, 1965. 6. LADYZHENSKAYA O. A. Boundary-value problems of mathematical physics. –Moscow: Nauka, 1973 (in Russian). 7. MIKHAJLOV V. P. Differential equations in partial derivatives.–Moscow: Nauka, 1983 (in Russian). 8. LIONS J.-L. Quelques methodes de resolution des problemes aus limites non linearies. –Paris: Dunod, 1969. 9. GHIDAGLIA J.-M., TEMAM R. Regularity of the solutions of second order evolution equations and their attractors // Ann. della Scuola Norm. Sup. Pisa. –1987. –Vol. 14. –P. 485–511. 10. BABIN A. V., VISHIK M. I. Attractors of evolution equations. –Amsterdam: North-Holland, 1992. 11. CHUESHOV I. D. On a certain system of equations with delay, occurring in aeroelasticity // J. of Soviet Mathematics. –1992. –Vol. 58. –P. 385–390. 12. CHUESHOV I. D., REZOUNENKO A. V. Global attractors for a class of retarded quasilinear partial differential equations // Math. Phys., Anal. Geometry. –1995. –Vol. 2. –# 3/4. –P. 363–383. 13. BOUTET DE MONVEL L., CHUESHOV I. D., REZOUNENKO A. V. Long-time behavior of strong solutions of retarded nonlinear P.D.E.s // Commun. Partial Dif. Eqs. –1997. –Vol. 22. –P. 1453–1474. 14. BOUTET DE MONVEL L., CHUESHOV I. D. Non-linear oscillations of a plate in a flow of gas // C. R. Acad. Sci. –Paris. –1996. –Ser. I. –Vol. 322. –P. 1001–1006.

~~Chapter 5~~

~~Theory of Functionals that Uniquely Determine Long-Time Dynamics~~

~~Contents~~

. . . . § 1 Concept of a Set of Determining Functionals ...... 285 . . . . § 2 Completeness Defect ...... 296 . . . . § 3 Estimates of Completeness Defect in Sobolev Spaces . . . . 306 . . . . § 4 Determining Functionals for Abstract Semilinear Parabolic Equations ...... 317 . . . . § 5 Determining Functionals for Reaction-Diffusion Systems . 328 . . . . § 6 Determining Functionals in the Problem of Nerve Impulse Transmission ...... 339 . . . . § 7 Determining Functionals for Second Order in Time Equations ...... 350 . . . . § 8 On Boundary Determining Functionals ...... 358

~~. . . . References ...... 361~~

The results presented in previous chapters show that in many cases the asymptotic behaviour of infinite-dimensional dissipative systems can be described by a finite-dimensional global attractor. However, a detailed study of the structure of attractor has been carried out only for a very limited number of problems. In this regard it is of importance to search for minimal (or close to minimal) sets of natural parameters of the problem that uniquely determine the long-time behaviour of a system. This problem was first discussed by Foias and Prodi [1] and by Ladyzhenskaya [2] for the 2D Navier-Stokes equations. They have proved that the long-time behaviour of solutions is completely determined by the dynamics of the first N Fourier modes if N is sufficiently large. Later on, similar results have been obtained for other parameters and equations. The concepts of determining nodes and determining local volume averages have been introduced. A general approach to the problem on the existence of a finite number of determining parameters has been discussed (see survey [3]). In this chapter we develop a general theory of determining functionals. This theory enables us, first, to cover all the results mentioned above from a unified point of view and, second, to suggest rather simple conditions under which a set of functionals on the phase space uniquely determines the asymptotic behaviour of the system by its values on the trajectories. The approach presented here relies on the concept of completeness defect of a set of functionals and involves some ideas and results from the approximation theory of infinite-dimensional spaces.

~~§ 1 Concept of a Set of Determining Functionals~~

Let us consider a nonautonomous differential equation in a real reflexive Banach space H of the type u d ==Fu(), t, t > 0, u u .(1.1) dt t = 0 0 W º {}³ Let be a class of solutions to (1.1) defined on the semiaxis R+ t : t 0 such ()ÎW > that for any ut there exists a point of time t0 0 such that ()Î (), ¥ ; Ç 2 (), ¥, ut Ct0 + H Lloc t0 + V ,(1.2) where VH is a reflexive Banach space which is continuously embedded into . Here- inafter Cab(), ; X is the space of strongly continuous functions on []ab, with the 2 (), ; . . values in XL and loc ab X has a similar meaning. The symbols H and V . £ . stand for the norms in the spaces H and V , H V . 286 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C h The following definition is based on the property established in [1] for the Fou- a rier modes of solutions to the 2D Navier-Stokes system with periodic boundary con- p t ditions. e Let L = {}l : j = 1,,¼ N be a set of continuous linear functionals on V . r j Then L is said to be a set of asymptotically ()VH,,W -determining func- 5 tionals (or elements) for problem (1.1) if for any two solutions uv, ÎW the condition t + 1 ()()t - ()()t 2 t = = ,,¼ limlj u lj v d 0 forj 1 N (1.3) t ® ¥ ò t implies that ()- () = limut vt H 0 .(1.4) t ® ¥ Thus, if L is a set of asymptotically determining functionals for problem (1.1), the asymptotic behaviour of a solution ut() is completely determined by the behaviour {}()() = ,,¼ , of a finite number of scalar values lj ut : j 12 N . Further, if no ambi- guity results, we will sometimes omit the spaces V , H , and W in the description of determining functionals.

~~Exercise 1.1 Show that condition (1.3) is equivalent to t + 1 2 lim []NL ()u()t - v()t dt = 0 , t ® ¥ ò t~~

~~where NL ()u is a seminorm in H defined by the equation~~

~~NL ()u = max lu(). l ÎL~~

Exercise 1.2 Let u1 and u2 be stationary (time-independent) solutions to W L = {}= ,,¼ problem (1.1) lying in the class . Let lj : j 1 N be a set of asymptotically determining functionals. Show that condition ()= () = ,,¼ , = lj u1 lj u2 for all j 12 N implies that u1 u2 .

~~The following theorem forms the basis for all assertions known to date on the existence of finite sets of asymptotically determining functionals.~~

Theorem 1.1. L = {}= ,,¼ Let lj : j 1 N be a family of continuous linear functionals V (), ´ on V . SSuuupposeppose that there exists a continuous function ut on H R+ with the values in R+ which possesses the following properties: a) there exist positive numbers as and such that V ()a, ³ × s Î , Î ut u for all uHtR+ ; (1.5) Concept of a Set of Determining Functionals 287

b) for any two solutions ut(), vt()Î W to problem (1.1) there exist (i) >y() a point of time t0 0 , (ii) a function t that is locally integrable [ , ¥) over the half-interval t0 and such that ta+ + gy = lim yt()d t > 0 (1.6) t ® ¥ ò t and ta+ + Gy = lim max {}0, -yt()dt < ¥ (1.7) t ® ¥ ò t for some a > 0 , and (iii) a positive constant C such that for all ³³ tst0 we have t V ()ytut()-vt(), t + ò ()×tV ()u()t - v()()tt , d £ s t £ V ()()- (), + × ()()t - ()()t 2 t us vs s Clmax j u lj v d . (1.8) ò j = 1,,¼ N s Then L is a set of asymptotically ()VH,,W -determining functionals for problem (1.1).

~~It is evident that the proof of this theorem follows from a version of Gronwall's lemma stated below.~~

Lemma 1.1. Let y()t and gt() be two functions that are locally integrable over [ , ¥) () some half-interval t0 . Assume that (1.6) and (1.7) hold and gt is nonnegative and possesses the property ta+ limg()t d t= 0 ,.a > 0 (1.9) t ® ¥ ò t Suppose that wt() is a nonnegative continuous function satisfying the inequality t t wt()+ ò yt()×tw()t d £ ws()+ ò g()t d t (1.10) s s ³³ ()® ® ¥ for all tst0 . Then wt 0 as t .

~~It should be noted that this version of Gronwall's lemma has been used by many authors (see the references in the survey [3]). 288 Theory of Functionals that Uniquely Determine Long-Time Dynamics~~

C h Proof. a Let us first show that equation (1.10) implies the inequality p t ìüt t ìüt e ïï ïï r wt() £ ws()exp íý- ys()sd + g()t exp íý- ys()sd dt (1.11) ïïò ò ïïò 5 îþs s îþt for all tst³³ . It follows from (1.10) that the function wt() is absolutely 0 \ continuous on any finite interval and therefore possesses a derivative w()t almost everywhere. Therewith, equation (1.10) gives us \ w()t + y()t wt() £ gt() (1.12) for almost all t . Multiplying this inequality by

~~ìüt ïï et()= expíý ys()sd , ïïò îþs we find that~~

d ()wt()et() £ gt()et() dt almost everywhere. Integration gives us equation (1.11). Let us choose the value s such that t + a {}-ys(), sG£ + GGº + ò max 0 d 1 ,(1.13)y t and t + a g + ys()sd ³ ,(1.14)ggº ò 2 y t t - t for all t ³ s . It is evident that if t ³³t s and k = , where []. is the in- a teger part of a number, then

~~t ka + t t òys()sd = ò ys()sd + ò ys()sd ³ t t ka+t ()k +1 a +t g g ³³ k - max {}-ys(), 0 ds k - ()G + 1 . 2 ò 2 ka + t Thus, for all t ³³t s Concept of a Set of Determining Functionals 289~~

~~t g æög ys()sd ³ ()t - t - 1 ++G . ò 2 a èø2 t Consequently, equation (1.11) gives us that~~

~~t ìüg ìüg wt() £ C ()Gg, ws()exp íý- ()ts- + gr()exp íý- ()t - t dt , îþ2a ò îþ2a s ìüg where C ()Gg, = expíý1 ++G . Therefore, îþ2~~

t ìüg lim wt() £ C ()Gg, × lim g()t expíý- ()t -t dt .(1.15) t ® ¥ t ® ¥ ò îþ2a s It is evident that t ìüg Gt(), s º g()t exp íý- ()t -t dt £ ò îþ2a s sk+ ()+1 a N ìüg £ å g()t exp íý- ()t -t dt , ò îþ2a k = 0 ska+ - - = ts ts where N a is the integer part of the number a . Therefore,

~~s + a sk+ ()+1 a N ìüg Gt(), s £ sup g()tt d × å exp íý- ()t - t dt = s ³ s ò ò îþ2a s k = 0 ska+ s + a sN+ ()+1 a ìüg ==supg()tt d exp íý- ()t -t dt s ³ s ò ò îþ2a s s~~

s + a g ìüg ()N +1 2a 2 = g × sup g()tt d × exp íý- ()ts- e - 1 . s ³ s ò îþ2a s ts- Since N = , this implies that a s + a lim Gt(), s £ Ca(), g × sup g()t d t (1.16) t ® ¥ s ³ s ò s 290 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C h for any s such that equations (1.13) and (1.14) hold. Hence, equations (1.15) a and (1.16) give us that p t s + a e r lim wt() £ C ()Gg,,a × sup g()tt d . t ® ¥ s ³ s ò 5 s If we tend s ® ¥ , then with the help of (1.9) we obtain limwt()= 0 . t ® ¥ This implies the assertion of Lemma 1.1.

In cases when problem (1.1) is the Cauchy problem for a quasilinear partial differential equation, we usually take some norm of the phase space as the function V ()ut, when we try to prove the existence of a finite set of asymptotically determining functionals. For example, the next assertion which follows from Theorem 1.1 is often used for parabolic problems.

Corollary 1.1. Let VH and be reflexive Banach spaces such that V is continuously and densely embedded into H . Assume that for any two solutions (), ()Î W u1 t u2 t to problem (1.1) we have t u ()t - u ()t 2 + yt()u ()t - u ()t 2 t £ 1 2 V ò 1 2 V d s t 2 £ u ()s - u ()s + Ku- u 2 t (1.17) 1 2 V ò 1 2 H d s ³³ y() for tst0 , where K is a constant and the function t depends on () () u1 t and u2 t in general and possesses properties (1.6) and (1.7). L = {}= ,,¼ Assume that the family lj : j 1 N on V possesses the property £ × ()+ e v H Clmax j v L v V (1.18) j = 1 ¼ N

~~for any vVÎ , where C and eL are positive constants depending onLL . Then is a set of asymptotically determining functionals for problem (1.1), provided ta+ 2 1 1 + -1 -1 eL~~

Proof. Using the obvious inequality æö ()ab+ 2 £d()1 + d a2 + 1 + 1 b2 ,,> 0 èød we find from equation (1.18) that 2 £ ()e+ d 2 2 + × ()2 v H 1 L v V Cd max lj v (1.20) j = 1 ¼ N for any d > 0 . Therefore, equation (1.17) implies that t æö u ()t - u ()t 2 + yt()- ()1 + d K e2 u ()t - u ()t 2 t £ 1 2 V ò èøL 1 2 V d s t £ u ()s - u ()s 2 + C []N ()u ()t - u ()t 2 t , 1 2 V ò L 1 2 d s N ()= () d > where L v max lj v . Consequently, if for some 0 the function j = 1 ¼ N 2 y()t = y()t - ()1 + d KeL possesses properties (1.6) and (1.7) with some constants gG and > 0 , then Theorem 1.1 is applicable. A simple verification shows that it is sufficient to require that equation (1.19) be fulfilled. Thus, Corollary 1.1 is proved.

~~Another variant of Corollary 1.1 useful for applications can be formulated as follows.~~

Corollary 1.2. Let VH and be reflexive Banach spaces such that V is continuously embedded into Hut . Assume that for any two solutions (), vt()ÎW to > ³³ problem (1.1) there exists a moment t0 0 such that for all tst0 the equation t ()- ()2 +£n ()t - ()t 2 t ut vt H ò u v V d s t £ ()- ()2 + ft()×t()t - ()t 2 us vs H ò u v H d (1.21) s holds. Here n >f0 and the positive function ()t is locally integrable [ , ¥) over the half-interval t0 and satisfies the relation ta+ lim 1 ft()d t £ R (1.22) t ® ¥ a ò t 292 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C > > () h for some a 0 , where the constant R 0 is independent of ut and a vt(). Let L = {}l : j = 1,,¼ N be a family of continuous linear func- p j t tionals on V possessing the property e r £ × ()+ e × w H CL max lj w L w V j = 1,,¼ N 5 for any wVÎ . Here CL and eL are positive constants. Then L is a set of asymptotically ()VH,,W -determining functionals for problem (1.1),

~~provided that eL < n¤ R .~~

Proof. Equation (1.20) implies that 2 ³ ()+ d -1 ××e-2 2 - × ()2 w V 1 L w H CL, d max lj w j = 1,,¼ N for any d > 0 . Therefore, (1.21) implies that t ()- ()2 + yt()×t()t - ()t 2 £ ut vt H ò u v H d s t £ ()- ()2 + n × ()()t - ()t 2 t us vs H CL, d max lj u v d , ò j = 1,,¼ N s -1 -2 where y()t = n()1 + d eL - f()t . Using (1.22) and applying Theorem 1.1 with V ()ut, = u 2 , we complete the proof of Corollary 1.2.

Other approaches of introduction of the concept of determining functionals are also possible. The definition below is an extension to a more general situation of the property proved by O.A. Ladyzhenskaya [2] for trajectories lying in the global attractor of the 2D Navier-Stokes equations. Let W be a class of solutions to problem (1.1) on the real axis R such that W Ì 2 ()-¥, ¥ ; L = {}= ,,¼ Lloc + V . A family lj : j 1 N of continuous linear functionals on VV is said to be a set of (), W -determining functionals (or elements) for problem (1.1) if for any two solutions uv, Î W the condition ()() = ()() = ,,¼ Î lj ut lj vt forj 1 N and almost alltR (1.23) implies that ut()º vt() .

~~It is easy to establish the following analogue of Theorem 1.1.~~

Theorem 1.2. L = {}= ,,¼ Let lj : j 1 N be a family of continuous linear functionals on V . Let W be a class of solutions to problem (1.1) on the real axis R such that Concept of a Set of Determining Functionals 293

W Ì ()¥, ¥ ; Ç 2 ()¥, ¥ ; C – + H Lloc – + V . (1.24) Assume that there exists a continuous function V ()ut, on H ´ R with the values in R which possesses the following properties: a) there exist positive numbers as and such that s V ()aut, ³ × u for alluHÎ ,t Î R ; (1.25) b) for any ut(), vt()Î W sup V ()ut()- vt(), t < ¥ ; (1.26) t Î R c) for any two solutions ut(), vt()Î W to problem (1.1) there exist (i) a function y()t locally integrable over the axis R with the properties ta+ - gy º limyt()d t> 0 (1.27) t ® -¥ ò t and ta+ - Gy º lim max {}0, -yt()dt < ¥ (1.28) t ® -¥ ò t for some a > 0 , and (ii) a positive constant C such that equation (1.8) holds for all ts³ . Then L is a set of ()V, W -determining functionals for problem (1.1).

~~Proof. It follows from (1.23), (1.8), and (1.11) that the function wt()= V ()ut()-vt(), t satisfies the inequality~~

~~ìüt ïï wt() £ ws()× exp íý- yt()d t (1.29) ïïò îþs for all ts³ . Using properties (1.27) and (1.28) it is easy to find that there exist * > > numbers s , a0 0 , and b0 0 such that~~

~~s2 yt() t³ × ()- - ££* ò d a0 s2 s1 b0 ,.s1 s2 s~~

~~s1 This equation and boundedness property (1.26) enable us to pass to the limit in (1.29) for fixed ts as ® -¥ and to obtain the required assertion.~~

~~Using Theorem 1.2 with V ()ut, = u 2 as above, we obtain the following assertion. 294 Theory of Functionals that Uniquely Determine Long-Time Dynamics~~

C h Corollary 1.3. a p Let VH and be reflexive Banach spaces such that V is continuously t embedded into H . Let W be a class of solutions to problem (1.1) on the e r real axis R possessing property (1.24) and such that () < ¥ ()Î W 5 sup ut H for allut . (1.30) tRÎ Assume that for any ut(), vt()Î W and for all real ts³ equation (1.21) holds with n >f0 and a positive function ()t locally integrable over the axis R and satisfying the condition sa+ 1 lim ft()d t£ R (1.31) s ® -¥ a ò s for some a > 0 . Here R > 0 is a constant independent of ut() and vt() . L = {}= ,,¼ Let lj : j 1 N be a family of continuous linear functionals on V possessing property (1.18) with eL < n¤ R . Then L is a set of asymptotically ()V, W -determining functionals for problem (1.1).

Proof. As in the proof of Corollary 1.2 equations (1.20) and (1.21) imply that t ()- () + yt() ()t - ()t 2 t £ ()- ()2 ut vt H ò u v H d us vs H s -1 -2 for all ts³yt , where ()=d n()1 + d eL - ft() and is an arbitrary positive number. Hence

~~ìüt ïï ut()- vt()2 £ us()- vs()2 expíý- yt()td (1.32) H H ïïò îþs for all ts³h . Using (1.31) it is easy to find that for any > 0 there exists Mh > 0 such that~~

~~s2 ft() t £ ()+ h ()- + ò d R s2 s1 a~~

~~s1 ££- for all s1 s2 Mh . This equation and boundedness property (1.30) enable us to pass to the limit as s ®e-¥ in (1.32), provided L < n¤ R , and to obtain the required assertion.~~

We now give one more general result on the finiteness of the number of determining functionals. This result does not use Lemma 1.1 and requires only the convergence of functionals on a certain sequence of moments of time. Concept of a Set of Determining Functionals 295

Theorem 1.3. Let VH and be reflexive Banach spaces such that V is continuously embedded into H . Assume that W is a class of solutions to problem (1.1) possessing property (1.2). AssumAssumumee that there exist constants CK, ³ 0 , ba>> << () 0 , and 0 q 1 such that for any pair of solutions u1 t and () W u2 t from we have u ()t - u ()t £ Cu()s - u ()s ,,,sts££+ b , (1.33) 1 2 V 1 2 V and u ()t - u ()t £ Ku()t - u ()t + qu()s - u ()s , s +a ££ts+b (1.34) 1 2 V 1 2 H 1 2 V for s large enough. Let L be a finite set of continuous linear functionals e < ()- -1 {} on V possessing property (1.18) with L 1 q K . Assume that tk is a ®a¥ ££-b sequence of positive numbers such that tk + and tk +1 tk . Assume that ()()- ()= Î L limlu1 tk u2 tk 0 ,,.l . (1.35) k ® ¥ Then u ()t - u ()t ® 0 as t ® + ¥ . (1.36) 1 2 V

~~It should be noted that relations like (1.33) and (1.34) can be obtained for a wide class of equations (see, e.g., Sections 1.9, 2.2, and 4.6).~~

Proof. ()= ()- () Let ut u1 t u2 t . Then equations (1.34) and (1.18) give us () £ ()+ ()() utk qL utk -1 Clmax j utk , V V j = ()- e -1 < where qL q 1 L K 1 . Therefore, after iterations we obtain that n () £ n () + × nk- ()() utn qL ut0 Cqå L max lj utk . V V j k = 1 Hence, equation (1.35) implies that ut() ® 0 as n ® ¥ . Therefore, (1.36) fol- n V lows from equation (1.33). Theorem 1.3 is proved.

Application of Corollaries 1.1–1.3 and Theorem 1.3 to the proof of finiteness of a set L of determining elements requires that the inequalities of the type (1.17) and (1.21), or (1.30) and (1.33), or (1.33) and (1.34), as well as (1.18) with the constant eL small enough be fulfilled. As the analysis of particular examples shows, the fulfilment of estimates (1.17), (1.21), (1.30), (1.31), (1.33), and (1.34) is mainly connected with the dissipativity properties of the system. Methods for obtaining them are rather well-developed (see Chapters 1 and 2 and the references therein) and in many cases the corresponding constants n , RK , , and q either are close to opti- 296 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C h mal or can be estimated explicitly in terms of the parameters of equations. There- a fore, the problem of description of finite families of functionals that asymptotically p t determine the dynamics of the process can be reduced to the study of sets of func- e tionals for which estimate (1.18) holds with e small enough. It is convenient to r L base this study on the concept of completeness defect of a family of functionals with 5 respect to a pair of spaces.

~~§ 2 Completeness Defect~~

Let VH and be reflexive Banach spaces such that V is continuously and densely embedded into H . The completeness defect of a set L of linear functionals onVH with respect to is defined as ìü eL ()VH, = sup íýw : wVÎ , lw()= 0 , l Î L , w £ 1 .(2.1) îþH V It should be noted that the finite dimensionality of LinL is not assumed here.

~~Exercise 2.1 Prove that the value eL ()VH, can also be defined by one of the following formulae: ìü eL ()VH, = sup íýw : wVÎ , lw()= 0 , w = 1 ,(2.2) îþH V~~

~~ìüw H eL ()VH, = sup íý : wVÎ , w ¹ 0, lw()= 0 ,(2.3) îþw V~~

~~ìü eL ()VH, = inf íýC : w £ Cw , wVÎ , lw()= 0 .(2.4) îþH V~~

~~L Ì L * Exercise 2.2 Let 1 2 be two sets in the space V of linear functionals onV . Show that eL ()eVH, ³ L ()VH, . 1 2~~

~~Exercise 2.3 Let L Ì V * and let L be a weakly closed span of the set L * e ()e, = (), in the space V . Show that L VH L VH .~~

~~The following fact explains the name of the value eL ()VH, . We remind that a set L of functionals on Vlw is said to be complete if the condition ()= 0 for all l ÎL implies that w = 0 .~~

~~Exercise 2.4 Show that for a set L of functionals on V to be complete it is~~

~~necessary and sufficient that eL ()VH, = 0 . Completeness Defect 297~~

~~The following assertion plays an important role in the construction of a set of determining functionals.~~

~~Theorem 2.1.~~

Let eL = eL ()VH, be the completeness defect of a set L of linear functionals on VH with respect to . Then there exists a constant CL > 0 such that ìü u £ e u + C × sup íýlu(): l Î L , l £ 1 (2.5) H L V L îþ* for any element uVÎ , where L is a weakly closed span of the set L in V *.

Proof. Let L ^ = {}vVÎ : lv()= 0 , l Î L (2.6) be the annihilator of L . If u ÎL ^ , then it is evident that lu()= 0 for all l ÎL . Therefore, equation (2.4) implies that £ e Î L^ u H L u V for allu , (2.7) i.e. for u ÎL^ equation (2.5) is valid. Assume that u ÏL^ . Since L^ is a subspace in V , it is easy to verify that there exists an element w ÎL^ such that ìü - ==(), L^ - ÎL^ uw distV u inf íýuv: v .(2.8) V îþV {}Ì L^ Indeed, let the sequence vn be such that º (), L^ = - dudistV lim uvn . n ® ¥ V {} It is clear that vn is a bounded sequence in V . Therefore, by virtue of the reflexivity of the space Vw , there exist an element from L^ and a subsequence {}v nk such that v weakly converges to wk as ® ¥ , i.e. for any functional fVÎ * the nk equation ()- = ()- fu w lim fu vn k ® ¥ k holds. It follows that ()- ££- × fu w lim uvn f df . k ® ¥ k V * * Therefore, we use the reflexivity of V once again to find that ìü uw- = sup íýfu()- w : fVÎ * , f = 1 £ d . V îþ* - ³ - = However, uwV d . Hence, uwV d . Thus, equation (2.8) holds. 298 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C h Equation (2.7) and the continuity of the embedding of VH into imply that a ££+e- + - p u H w H uwH L w V Cu wV . t e It is clear that r £ + - w V u V uwV . 5 Therefore, £ e + ()e + - u H L u V L C uwV .(2.9)

~~Let us now prove that there exists a continuous linear functional l0 on the space V possessing the properties ()- = - = ()= ÎL^ l0 uw uwV ,,l0 * 1 l0 v 0 for.(2.10)v~~

~~To do that, we define the functional l0 by the formula~~

~~()= - = + ()- l0 m au wV ,,mvauw on the subspace ìü Mmvauw= íý= + ()- : v ÎL^ , a Î R . îþ~~

()= ÎL^ It is clear that l0 is a linear functional on Ml and 0 m 0 for m . Let us calculate its norm. Evidently m º vauw+ ()- = auw× - + 1 v ,.a ¹ 0 V V a V Since wa- -1 v Î L^ , equation (2.8) implies that

~~³ × - = () = + ()- ¹ m V auwV l0 m ,,.mvauw a 0 Consequently, for any mMÎ~~

~~()£ l0 m m V .~~

This implies that l has a unit norm as a functional on M . By virtue of the Hahn-Ba- 0 nach theorem the functional l0 can be extended on V without increase of the norm. Therefore, there exists a functional l on V possessing properties (2.10). Therewith 0 L L ÏL l0 lies in a weakly closed span of the set . Indeed, if l0 , then using the reflexivity of Vl and reasoning as in the construction of the functional 0 it is easy to Î ()¹ ()= verify that there exists an element xV such that l0 x 0 and lx 0 for all l ÎL . It is impossible due to (2.10) . In order to complete the proof of Theorem 2.1 we use equations (2.9) and (2.10). As a result, we obtain that £ e + ()e + ()- u H L u V L C l0 uw.

~~However, l Î L ,l ()uw- = l ()u , and l = 1 . Therefore, equation (2.5) 0 0 0 0 * holds. Theorem 2.1 is proved. Completeness Defect 299~~

~~L = {}= ,,¼ * Exercise 2.5 Assume that lj : j 1 N is a finite set in V . Show that there exists a constant CL such that £ e (), + () u H L VH u V CL max lj u (2.11) j = 1,,¼ N for all uVÎ .~~

In particular, if the hypotheses of Corollaries 1.1 and 1.3 hold, then Theorem 2.1 and equation (2.11) enable us to get rid of assumption (1.18) by replacing it with the corresponding assumption on the smallness of the completeness defect eL ()VH, .

~~The following assertion provides a way of calculating the completeness defect when we are dealing with Hilbert spaces.~~

Theorem 2.2. Let VH and be separable Hilbert spaces such that V is compactly and densely embedded into HK. Let be a selfadjoint positive compact operator in the space V defined by the equality (), = (), , Î Ku v V uvH ,,.uv V. Then the completeness defect of a set L of functionals on V can be evaluated by the formula e (), = m() L VH max PL KPL , (2.12) where PL is the orthoprojector in the space V onto the annihilator L^ = {}vVÎ : lv()= 0 , l Î L m () and max S is the maximal eigenvalue of the operator S .

Proof. It follows from definition (2.1) that e (), = {}Î L VH sup u H : uBL = L^ Ç {}£ L^ where BL v : v V 1 is the unit ball in . Due to the compactness of the embedding of VH into , the set BL is compact in H . Therefore, there exists Î an element u0 BL such that e ()VH, 2 ==u 2 ()Ku , u º m . L 0 H 0 0 V (), Therewith u0 is the maximum point of the function Ku u V on the set BL . Hence, for any v Î L^ and s Î R1 we have ()Ku()+ sv , u + sv 0 0 V £ m º (), Ku0 u0 . u + sv 2 V 0 V 300 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C h It follows that a ()Ku()+ sv , u + sv - m u + sv 2 £ 0 . p 0 0 V 0 V t e It is also clear that u = 1 . Therefore, r 0 V s2 {}()Kv, v - m v 2 + 2 sKu{}(), v - m()u , v £ 0 5 V 0 V 0 V for all s Î R . This implies that ()mKu , v - ()u , v £ 0 0 0 V for any v ÎL^ . If we take -v instead of v in this equation, then we obtain the op- posite inequality. Therefore, ()m, - (), = Î L^ Ku0 v u0 v 0 ,.v Consequently, = m PL KPL u0 u0 , i.e. m ==()Ku , u u 2 is an eigenvalue of the operator P KP . It is evident 0 0 V 0 H L L that this eigenvalue is maximal. Thus, Theorem 2.2 is proved.

Corollary 2.1. {} Assume that the hypotheses of Theorem 2.2 hold. Let ej be an orthonormal basis in the space V that consists of eigenvectors of the operator K : Ke = m e ,,,.()e , e =md ³³m ¼ m® 0 j j j i j V ij 1 2 N Then the completeness defect of the system of functionals L = {}l Î V * : l ()v = ()ve, : j = 12,,¼ ,N j j j V e ()m, = is given by the formula L VH N +1 .

~~To prove this assertion, it is just sufficient to note that PL is the orthoprojector {}³ + onto the closure of the span of elements ej : jN1 and that PL commutes with K .~~

Exercise 2.6 Let A be a positive operator with discrete spectrum in the space H : Ae =ll e ,,,,££l ¼ l® + ¥ ()e , e = d k k k 1 2 N k j H kj = ()s Î and let s DA , s R , be a scale of spaces generated by the operator A (see Section 2.1). Assume that L = {}l : l ()v = ()ve, : j = 12,,¼ ,N .(2.13) j j j H e ()l , =s-()s - s > Prove that L s s N +1 for all s .

~~It should be noted that the functionals in Exercise 2.6 are often called modes. Completeness Defect 301~~

~~Let us give several more facts on general properties of the completeness defect.~~

Theorem 2.3. Assume that the hypotheses of Theorem 2.2 hold. Assume that L is a set of linear functionals on V and KL is a family of linear bounded operators RVH that map into and are such that Ru= 0 for all u ÎL^ . Let ìü e H()R = sup íýuRu- : u £ 1 (2.14) V îþH V be the global approximation error in H arising from the approximation of elements vVÎ by elements Rv. Then

~~ìüH eL ()VH; = min íýe ()R : R ÎKL . (2.15) îþV~~

~~Proof.~~

Let R ÎK L . Equation (2.14) implies that - £ H() Î uRuH eV R u V ,.uV ÎL^ £ H() e (); £ H() Therefore, for u we have u H eV R u V , i.e. L VH eV R for all ÎK ÎK R L . Let us show that there exists an operator R0 L such that e (); = H() L VH eV R0 . Equation (2.12) implies that ìü 12¤ 12¤ eL ()VH; ==K PL (); sup íýK PL u : u £ 1 , LV V îþV V L^ 12¤ where PL is the orthoprojector in the space V onto and K PL LVV(), is the 12¤ norm of the operator K PL in the space LVV(), of bounded linear operators in V . Therefore, the definition of the operator K implies that

~~ìüH eL ()VH; ==sup íýPL u : u £ 1 e ()IP- L .(2.16) îþH V V~~

~~It is evident that the orthoprojector QL = IP- L belongs to KL (it projects onto the subspace that is orthogonal to L^ in V ). Theorem 2.3 is proved.~~

L = {}= ,,¼ Exercise 2.7 Assume that lj : j 1 N is a finite set. Show that the family KL consists of finite-dimensional operators R of the form N = ()j Î Ruå lj u j ,,uV j = 1 {}j = ,,¼ where j : j 1 N is an arbitrary collection of elements of the space V (they do not need to be distinct). How should the {}j choice of elements j be made for the operator QL from the proof of Theorem 2.3? 302 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C h Theorem 2.3 will be used further (see Section 3) to obtain upper estimates of a the completeness defect for some specific sets of functionals. The simplest situation p t is presented in the following example. e r 5 Example 2.1 = 2 (), = ()2 Ç 1 (), s(), Let HL0 l and let VHH0 0 l . As usual, here H 0 l is s (), ¥ (), the Sobolev space of the order sH and 0 0 l is the closure of the set C0 0 l in Hs()0, l . We define the norms in HV and by the equalities l 2 = 2 º ()()2 2 = ¢¢ 2 u H u ò ux dx ,.u V u 0 = ¤ = = , ¼, - Let hlN , xj jh , j 1 N 1 . Consider a set of functionals L = {}()= () = , ¼, - lu uxj : j 1 N 1 on VR . Assume that is a transformation that maps a function uVÎ into its linear interpolating spline N -1 æö sx()= ux()cx - j . å j èøh j = 1 Here c()x = 1 - x for x £ 1 and c()x = 0 for x > 1 . We apply Theo- rem 2.3 and obtain ìü eL ()VH; £ sup íýus- : uVÎ , u¢¢ £ 1 . îþ We use an easy verifiable equation

~~x xj +1 x ux()- sx()= -t1 d d xu¢¢()y dy ,,xxÎ [], x h ò ò ò j j +1 t xj xj to obtain the estimate 2 us- £ h u¢¢ . 3 2 This implies that eL ()HV; £ h ¤ 3 .~~

~~The assertion on the interdependence of the completeness defect eL and the Kol- mogorov N -width made below enables us to obtain effective lower estimates~~

foreL ()VH; . Let VH and be separable Hilbert spaces such that V is continuously and densely embedded into HN . Then the Kolmogorov -width of the embedding of VH into is defined by the relation Completeness Defect 303

ìü == ()VH; inf íýeH ()F : F Î ,(2.17) N N îþV N where N is the family of all NFV -dimensional subspaces of the space and ìü eH ()F = sup íýdist ()vF, : v £ 1 V îþH V is the global error of approximation of elements vVÎ in H by elements of the subspace F . Here ìü dist ()vF, = inf íývf- : fFÎ . H îþH In other words, the Kolmogorov N -width N of the embedding of VH into is the minimal possible global error of approximation of elements of VH in by elements of some N -dimensional subspace.

~~Theorem 2.4. Let VH and be separable Hilbert spaces such that V is continuously and densely embedded into H . Then~~

ìü* ()VH; ==min íý eL ()VH; : L Ì V ; dim Lin L = N m + , (2.18) N îþN 1 {}m where j is the nonincreasing sequence of eigenvalues of the operator K (), = (), defined by the equality uvH Ku v V .

The proof of the theorem is based on the lemma given below as well as on the fact that ()uv, = ()Ku, v , where KV is a compact positive operator in the space H V ¥ (see Theorem 2.2). Further the notation {}e stands for the proper basis of the j j = 1 m operator KV in the space while the notation j stands for the corresponding eigenvalues: Ke =mm e ,,,.³³m ¼ m® 0 ()e , e = d j j j 1 2 n i j V ij ìü It is evident that 1 e is an orthonormalized basis in the space H . íým i îþi

~~Lemma 2.1. Assume that the hypotheses of Theorem 2.4 hold. Then~~

~~ìü* min íýeL ()VH; : L Ì V ,Lindim L = N = m + .(2.19) îþN 1 304 Theory of Functionals that Uniquely Determine Long-Time Dynamics~~

C h Proof. a By virtue of Corollary 2.1 it is sufficient to verify that p t e ()m; ³ e L VH N +1 r L = {}= ,,¼ for all lj : j 1 N , where lj are linearly independent functionals 5 onV . Definition (2.1) implies that ¥ []e ()VH; 2 ³mu 2 = ()ue, 2 (2.20) L H å j j V j = 1 Î = ()= = ,,¼ for all uV such that u V 1 and lj u 0 , j 1 N . Let us substitute in (2.20) the vector N +1 = ucå j ej , j = 1 ()= = ,,¼ where the constants cj are choosen such that lj u 0 for j 1 N and = u V 1 . Therewith equation (2.20) implies that N +1 N +1 []e (); 2 ³³m 2 m 2 == m2 m L VH å j cj N +1 å cj N + 1 u V N +1 . j = 1 j = 1 Thus, Lemma 2.1 is proved.

= {}e We now prove that N min L . Let us use equation (2.16) ìü eL ()VH; = sup íýuQ- L u : u £ 1 .(2.21) îþH V ^ Here QL is the orthoprojector onto the subspace QL V orthogonal to L in V . It is evident that QL V is isomorphic to Lin L . Therefore, dim QL V = N . Hence, equation (2.21) gives us that ìü eL ()VH; ³³sup íýdist()uQ, L V : u £ 1 (2.22) îþH V N for all L ÎV * such that dim Lin L = N . Conversely, let FN be an -dimensional {}= ,,¼ subspace in Vf and let j : j 1 N be a orthonormalized basis in the space H . Assume that L = {}l : l ()v = ()f , v : j = 1,,¼ N . F j j j H

Let QHF, be the orthoprojector in the space HF onto . It is clear that N Q uuf= (), f . HF, å j H j j = 1 ÎL ^ = Therefore, if u F , then QHF, u 0 . It is clear that QHF, is a bounded operator from VH into . Using Theorem 2.3 we find that Completeness Defect 305

H ìü eL ()VH; £ e ()Q , = sup íýuQ- , u : u £ 1 . F V HF îþHF H V However, uQ- u = dist ()uF, . Hence, HF, H H H min {}eL ££eL ()VH; e ()F (2.23) F V for any NFV -dimensional subspace in . Equations (2.22) and (2.23) imply that

~~ìü* ()VH; = min íý eL ()VH; : L Ì V , dimLinL = N . N îþ This equation together with Lemma 2.1 completes the proof of Theorem 2.4.~~

Exercise 2.8 Let Vk and Hk be reflexive Banach spaces such that Vk is con- L tinuously and densely embedded into Hk , let k be a set of linear = , functionals on Vk , k 12 . Assume that LL= È L Ì ()´ * 1 2 V1 V2 , where

~~ìü* Lk = íýlVÎ ()´ V : lv(), v = lv(), l Î L ,.k = 12, îþ1 2 1 2 k k Prove that ìü eL ()eV ´ V , H ´ H = max íýL ()eV , H , L ()V , H . 1 2 1 2 îþ1 1 1 2 2 2~~

Exercise 2.9 Use Lemma 2.1 and Corollary 2.1 to calculate the Kolmogorov = ()s = ()s N-width of the embedding of the space s DA into s DA for s > s , where A is a positive operator with discrete spectrum. (), = 2 × []p ()+ -2 Exercise 2.10 Show that in Example 2.1 N VH l N 1 . -2 2 2 Prove that p h ££eL ()VH; h ¤ 3 .

~~Exercise 2.11 Assume that there are three reflexive Banach spaces V Ì ÌÌWH such that all embeddings are dense and continuous.~~

~~LetL be a set of functionals on W . Prove that eL ()£VH, £ eL ()eVW, × L ()WH, (Hint: see (2.3)).~~

~~Exercise 2.12 In addition to the hypotheses of Exercise 2.11, assume that the inequality £ q × 1 - q Î u W aq u H u V , uV ,~~

~~holds for some constants aq >q0 and Î ()01, . Show that 1 1 -1 q 1-q []aq eL ()VW, ££eL ()VH, []aq eL ()WH, . 306 Theory of Functionals that Uniquely Determine Long-Time Dynamics~~

C h § 3 Estimates of Completeness Defect a p in Sobolev Spaces t e r 5 In this section we consider several families of functionals on Sobolev spaces that are important from the point of view of applications. We also give estimates of the corresponding completeness defects. The exposition is quite brief here. We recommend that the reader who does not master the theory of Sobolev spaces just get acquain- ted with the statements of Theorems 3.1 and 3.2 and the results of Examples 3.1 and 3.2 and Exercises 3.2–3.6. We remind some definitions (see, e.g., the book by Lions-Magenes [4]). Let W be a domain in Rn . The Sobolev space H m()W of the order mm ()= 012,,,¼ is a set of functions ìü H m()W = íýfLÎ 2 ()W : D j fx()Î L2 ()W , jm£ , îþ = (),,¼ = ,,,¼ = ++¼ where jj1 jn ,jk 012 ,jj1 jn and ¶ j f D j fx()= .(3.1) j j j ¶ 1 × ¶ 2 ¼¶ n x1 x2 xn The space H m()W is a separable Hilbert space with the inner product

(), = D j × D j uvm å ò u v dx . jm£ W m()W m()W Further we also use the space H0 which is the closure (in H ) of the set ¥()W W C0 of infinitely differentiable functions with compact support in and the space H s ()Rn which is defined as follows:

n ìün 2 s 2 H s()R = íýux()Î L2()R : () 1 + y u ()y dy º u 2 < ¥ , îþò s Rn where s ³ 0 , u ()y is the Fourier transform of the function ux() , u ()y = ò eixy ux()dx , Rn 2 = 2 ++¼ 2 = ++¼ y y1 y2 , and xy x1 y1 xn yn . Evidently this definition coincides with the previous one for natural s and W = Rn .

Exercise 3.1 Show that the norms in the spaces Hs()Rn possess the property q 1 - q u ()q £ u × u ,,s()q = qs + ()1 - q s s s1 s2 1 2 ££q , ³ 0 1 ,.s1 s2 0 Estimates of Completeness Defect in Sobolev Spaces 307

We can also define the space Hs()W as restriction (to W ) of functions from Hs()Rn with the norm ìü s n u , W = inf íýv n : vx()= ux() in W , vHÎ ()R s îþs, R s ()W ¥()W s()W s()W and the space H0 as the closure of the set C0 in H . The spaces H s ()W and H0 are separable Hilbert spaces. More detailed information on the Sobolev spaces can be found in textbooks on the theory of such spaces (see, e.g., [4], [5]). The following version of the Sobolev integral representation will be used further.

Lemma 3.1. Let W be a domain in Rn and let l()x be a function from L¥()Rn such that supp lÌÌl W,.ò ()x dx = 1 (3.2) Rn Assume that W is a star-like domain with respect to the support supp l of the function l . This means that for any point x Î W the cone = {}= t + ()- t ; ££t , Î l Vx z x 1 y 0 1 y supp (3.3) belongs to the domain W . Then for any function ux()Î Hm()W the representation

()= (); + m ()- a (), D a () ux Pm - 1 xu å a! ò xyKx y uydy (3.4) a = m Vx is valid, where (), = 1 l()()- a D a () Pm -1 xu å a! ò y xy uydy , (3.5) a < m W a a aa=a!a(),,¼a = !¼a ! a = 1 ¼ n 1 n ,,,1 n z z1 zn 1 æöyx- Kx(), y = s-n - 1 l x + ds . (3.6) ò èøs 0

Proof. If we multiply Taylor’s formula ()- a ()= xy D a ()+ ux å a! uy a < m 1 ()- a + xy m -1 D a ()+ ()- m å a! òs ux sy x ds a = m 0 308 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C l () h by y and integrate it over y , then after introducing a new variable a zxsyx= + ()- we obtain the assertion of the lemma. p t e Integral representation (3.4) enables us to obtain the following generalization of the r Poincaré inequality. 5 Lemma 3.2. Let the hypotheses of Lemma 3.1 be valid for a bounded domain W ÌlRn and for a function ()x . Then for any function ux()Î H1()W the inequality s n n + 1 uu- áñl £ d l ¥ × Ñu (3.7) L2 ()W n L ()W L2()W

~~is valid, where áñu =sl()x ux() x , is the surface measure of l òW d n the unit sphere in Rn and d = diam W º sup {}xy- : xy, Î W .~~

Proof. We use formula (3.4) for m = 1 : n ¶ ()= áñ+ (), ()- × u () ux u l å òKx y xj yj ¶ y dy .(3.8) yj j = 1 Vx It is clear that l ()x + ()1¤ s ()yx- = 0 when s-1 xy- ³ d º diam W . There- fore, 1 -n - æö1 Kx(), y = s 1 l x + ()yx- ds . ò èøs d-1 xy- Consequently, d dn Kx(), y £ ,ddlnº (), = l . (3.9) xy- n n L¥()W Thus, it follows from (3.8) that Ñ () ()-dáñ ££uy ux u l n - dy ò xy- 1 W

æöÑuy()2 12æö12 £ d dy ç÷n - dy ç÷n - . (3.10) èøò xy- 1 èøò xy- 1 W W ()= {}- £ Let Br x y : xy r . Then it is evident that dy ££dy s × n - n - n d ,(3.11) ò xy- 1 ò xy- 1 W () Bd x Estimates of Completeness Defect in Sobolev Spaces 309

n where sn is the surface measure of the unit sphere in R . Therefore, equation (3.10) implies that Ñ ()2 ()- áñ2 £ d2 ××s uy ux u l n d n - dy . ò xy- 1 W After integration with respect to x and using (3.11) we obtain (3.7). Lemma 3.2 is proved.

Lemma 3.3. Assume that the hypotheses of Lemma 3.1 are valid for a bounded do- n main W from R and for a function l()x . Let mn = []n¤ 2 + 1 , where []. is a sign of the integer part of a number. Then for any function m ux()Î H n ()W we have the inequality

()- (); £ max ux Pm -1 xu x Î W n n c mn + m £ n d 2 l n D a u , n L¥()W å a! L2()W (3.12) a = mn (); = []s ¤ ()- n 12 where Pm -1 xu is defined by formula (3.5), cn n 2 mn , n sn is the surface measure of the unit sphere in R , and d = diam W .

~~Proof. Using (3.4) and (3.9) we find that~~

~~mn m ux()- P ()xu; ££ xy- n Kx(), y × D a uy()dy mn -1 å a! ò a = mn Vx~~

~~d m m - n ££n - n × D a () å a! ò xy uy dy a = mn W~~

~~12 d mn æö2 ()m - n £ D a u × ç÷xy- n dy . å a! L2()W èøò a = mn W~~

As above, we obtain that d ()- n - n - - n - 2 mn ££s 2 mn 1 s 2 mn × ()- n -1 ò xy dy n ò r dr n d 2 mn . W 0 Thus, equation (3.12) is valid. Lemma 3.3 is proved. 310 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C h Lemma 3.4. a p Assume that the hypotheses of Lemma 3.3 hold. Then for any function mn t ux()Î H ()W and for any x* Î W the inequality e r æö12 uux- () º ux()- ux()2 dx £ 5 * 2 ç÷ò * L ()W èø W

~~mn 1 £ C ××()dn l d j D a u n L¥()W å å a! L2()W (3.13) j = 1 a = j~~

~~is valid, where Cn is a constant that depends on n only and d is the diameter of the domain W .~~

~~Proof. It is evident that n2 uux- () £ uu- áñl + d áñu l - ux(),(3.14) * L2()W L2()W * where áñ = l() () u l ò y uydy . W (), The structure of the polynomial Pm -1 xu implies that~~

~~áñu l - ux() £~~

~~1 a a £ ux()- P ()xu, + l()y ()xy- D uy()dy £ mn -1 å a! ò 1 ££a mn - 1 W a d a £ ux()- P ()xu, + l × D u 2 mn -1 å a! L2()W L ()W 1 ££a mn -1~~

~~for all x Î W . Therefore, estimate (3.13) follows from (3.14) and Lemmata 3.2 and 3.3. Lemma 3.4 is proved.~~

These lemmata enable us to estimate the completeness defect of two families of functionals that are important from the point of view of applications. We consider these families of functionals on the Sobolev spaces in the case when the domain is strongly Lipschitzian, i.e. the domain W Ì Rn possesses the property: for every x Î ¶W there exists a vicinity U such that Ç W = {}= (),,¼ < (),,¼ U xx1 xn : xn fx1 xn -1 Estimates of Completeness Defect in Sobolev Spaces 311 in some system of Cartesian coordinates, where fx() is a Lipschitzian function. For strongly Lipschitzian domains the space Hs()W consists of restrictions to W of functions from Hs()Rn , s > 0 (see [5] or [6]).

Theorem 3.1. Assume that a bounded strongly Lipschitzian domain W inRn can be {}W = ,,¼ , divided into subdomains j : j 12 N such that N WW= W ÇÆW = ¹ È j , i j for ij. (3.15) j = 1 l () Here the bar stands for the closure of a set. Assume that j x is a function ¥()W in L j such that l ÌWÌl() = supp j j , ò j x dx 1 (3.16) W j W l L and j is a star-like domain with respect to supp j . We define the set of generalized local volume averages corresponding to the collection T = {}()W ; l = ,,¼ , j j : j 12 N as the family of functionals ìü L = íýl ()u = l ()x ux()dx , j = 12,,¼ ,N . (3.17) îþj ò j W j Then the estimate

~~ì s - s ïC ()Ln, s ()d , s > 1, 0 ££s s , e ()s()W , s ()W £ L H H í - s (3.18) ï L1 s × s - s £££s ¹ îCn d , 0 s 1, s 0 ,~~

~~ìü L = n l = ,,¼ , = holds, where max íýdj j ¥ : j 12 N ,,,ddmax j , îþL ()W j º W = {}- , Î W dj diam sup xy: xy j ,~~

~~C ()n, s and Cn are constants.~~

~~Proof.~~

Let us define the interpolation operator R T for the collection T by the formula ()R ()==() l() () Î W = ,,¼ , T u x lj u ò j x uxdx ,,x j j 12 N . W j 312 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C h It is easy to check that a - R £ L Î 2()W p u T u Cn u ,.uL L2()W L2()W t e It further follows from Lemma 3.2 that r sn n + 5 ul- ()u £ d 1 l Ñu . j 2()W n j j ¥()W L2()W L j L j j This implies the estimate sn u - R T u £ d L u . L2()W n H1()W Using the fact (see [4, 6]) that - u £ Cu1 s × u s ,,0 ££s 1 Hs()W L2()W H1()W and the interpolation theorem for operators [4] we find that s s 1 - s æön uR- u £ ()Cn L dL u T L2()W èøn H s()W for all 0 ££s 1 . Consequently, Theorem 2.3 gives the equation s 2 s eL ()H ()W ; L ()W £ Cn Ld ,.(3.19)0 ££s 1 Using the result of Exercise 2.12 and the interpolation inequalities (see, e.g., [4,6]) £ q × 1 - q ()q = q + ()- q ££q u s()q ()W Cu s u s ,,,(3.20)s s1 s2 1 0 1 H H 1()W H 2()W it is easy to obtain equation (3.18) from (3.19).

~~Let us illustrate this theorem by the following example.~~

Example 3.1 Let W = ()0, l n be a cube in Rn with the edge of the length l . We construct a collection T which defines local volume averages in the following way. LetK = ()01, n be the standard unit cube in Rn and let w be a measurable set in K with the positive Lebesgue measure, mesw > 0 . We define the function lw(), x on K by the formula ì []w -1 Î w lw(), x = í mes , x , î 0 , xKÎ \ w . Assume that ìü xi W = hjK× ()+ º íýxx==(),,¼ xn : j << j + 1, i 12,,¼n , , j îþ1 i h i

~~æö l ()x = 1 lw, x - j ,,x Î W j hn èøh j Estimates of Completeness Defect in Sobolev Spaces 313~~

= (),,¼ = ,,¼ , - = ¤ for any multi-index jj1 jn , where jn 01 N 1 , hlN . It is clear that the hypotheses of Theorem 3.1 are valid for the collection T = {}W , l j j . Moreover, nn2 d ==diam W n × h , L = j j mesw and hence in this case we have - s æöh s e ()Hs()W ; Hs()W £ C (3.21) L n, s èømes w for the set L of functionals of the form (3.17) when s ³ 1 and 0 ££s s . It should be noted that in this case the number of functionals in the set L is n equal to NL = N . Thus, estimate (3.21) can be rewritten as - s - s æös æös e ()s()W , s()W £ l × 1 n L H H Cn, s ç÷w ç÷N . èømes èøL

However, one can show (see, e.g., [6]) that the Kolmogorov NL -width of the em- s s bedding of H ()W into H ()W has the same order in NL , i.e. - s æös ()s()W , s()W = 1 n N H H c0 ç÷ . L èøNL Thus, it follows from Theorem 2.4 that local volume averages have a completeness defect that is close (when the number of functionals is fixed) to the minimal. In the example under consideration this fact yields a double inequality s - s ££se ()s()W , s()W s - s < c1 h L H H c2 h , s , (3.22) nw W where c1 and c2 are positive constants that may depend on s , , , and . Similar relations are valid for domains of a more general type.

~~Another important example of functionals is given in the following assertion.~~

~~Theorem 3.2.~~

Assume the hypotheses of Theorem 3.1. Let us choose a point xj (called W m()W a node) in every set j and define a set of functionals on H , m = []n¤ 2 + 1 , by ìü L = íýl ()u = ux(): x Î W , j = 1,,¼ N . (3.23) îþj j j j Then for all sm³ and 0 ££s s the estimate s s s - s eL ()H ()W ; H ()W £ C()n, s ()d L (3.24) is valid for the completeness defect of this set of functionals, where 314 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C ìü h = = ,,¼ , = W a ddmax íýj : j 12 N ,,,dj diam , p îþ t e ìün r L = max íýd l : j = 12,,¼ ,N . îþj j L¥()W 5 Proof. Î m ()W ()==() = ,,¼ , Let uH and let lj u uxj 0 ,j 12 N . Then using WW= = (3.13) for j and x* xj we obtain that m £ æön l l u 2()W Cn èødj j ¥ ådj u l, W , L j L ()W j j l = 1 where = 1 D a u l, W å u 2()W . j a! L j a = l It follows that m u £ C()Ln ×× dl u L2()W å Hl ()W l = 1 Î m()W ()==() = ,,¼ , for all uH such that lj u uxj 0 , j 12 N . Using interpolation inequality (3.20) we find that l m 1 - l m lm u £ Cn ××L d u × u .(3.25) L2()W å L2()W Hm()W l = 1 By virtue of the inequality p yq £ x + ,,,1 + 1 = 1 , ³ 0 xy p q p q xy we get

~~l l m l 1 - 1 - æö m L ××dl u m ×u lm = u m × ç÷dm L l u £ L2 ()W Hm()W L2 èøHm~~

~~m -m m £ ()d1 - l ml- u + l × d l d mL l u m L2()W m Hm()W~~

for l =d12,,¼ ,m -1 and for all > 0 . We substitute these inequalities in (3.25) to obtain that m - 1 m l u £ C ()d1 - ml- × u + L2()W n å m L2()W l = 1 m - 1 m m æöl - + C ç÷ d l L l + L d m u . (3.26) n èøå m Hm()W l = 1 Estimates of Completeness Defect in Sobolev Spaces 315

We choose ddn= (), m such that m - 1 m æöl 1 C 1 - d ml- £ . n å èøm 2 l = 1 Then equation (3.26) gives us that m m l u £ C()n × L l × dm u L2()W å m Hm()W l = 1 Î m()W ()= = ,,¼ , for all uH such that uxj 0 , j 12 N . Hence, the estimate m m l e ()Hm()W ; L2()W £ C()n dm × L l L å m l = 1 is valid. Since L ³s1 , this implies inequality (3.24) for = 0 and sm== = []n¤ 2 + 1 . As in Theorem 3.1 further arguments rely on Lemma 2.1 and interpolation inequalities (3.20). Theorem 3.2 is proved.

Example 3.2 Î W We return to the case described in Example 3.1. Let us choose nodes xj j and assume that w = K . Then for a set L of functionals of the form (3.23) we have e ()s()W s()W £ s - s L H ; H Cn, s h ³ []n¤ + ££s for all s 2 1 , 0 s and for any location of the nodes xj inside the W j . In the case under consideration double estimate (3.22) is preserved.

~~In the exercises below several one-dimensional situations are given.~~

~~Exercise 3.2 Prove that -1 æöp N 2 2 e ()H1()0, l ; L2()0, l ==()H1()0, l ; L2()0, l 1 + L N èøl for l ìüjp L = íýl c ()u ==ux()cos x dx , j 01,,¼ ,N - 1 . îþj ò l 0~~

~~Exercise 3.3 Verify that~~

1 æöpN 2 - e ()H1 ()0, l ; L2()0, l ==()H1()0, l ; L2()0, l 1 + 2 L 0 N -1 0 èøl for l ìüjp x L = íýl s ()u ==ux()sin dxj; 12,,¼ ,N - 1 . îþj ò l 0 316 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C h Exercise 3.4 Let a 1 p H ()0, l = {}uHÎ 1()0, l : u()0 = ul() . t per e Show that r e ()1 (), ; 2(), = 5 L Hper 0 l L 0 l 1 æö2 p N 2 - == ()H1 ()0, l ; L2 ()0, l 1 + 2 , 2 N - 1 per èøl L c s = ,,¼ , - where consists of functionals l2 k and l2 k for k 01 N 1 c s (the functionals lj and lj are defined in Exercises 3.2 and 3.3). Exercise 3.5 Consider the functionals h 1 l ()u = ux()t+ t d , j h ò j 0

x = jh,,h = l j = 01,,¼ ,N -1 , j N 2(), on the space L 0 l . Assume that an interpolation operator Rh Î 2(), () maps an element uL0 l into a step-function equal to lj u [], on the segment xj xj +1 . Show that uR- u £ hu¢ . h L2()0, l L2()0, l Prove the estimate

~~h 1 2 ££eL ()H ()0, l ; L ()0, l h . p2 + h2~~

~~Exercise 3.6 Consider a set L of functionals~~

l ()u = ux(),,,x = jh h = l j = 01,,¼ ,N -1 , j j j N 1(), on the space H 0 l . Assume that an interpolation operator Rh Î 1 (), () maps an element uH0 l into a step-function equal to lj u [], on the segment xj xj +1 . Show that - £ h ¢ uRhu u 2 . L2()0, l 2 L ()0, l Prove the estimate

~~h 1 2 h ££eL ()H ()0, l ; L ()0, l . p2 + h2 2 Determining Functionals for Abstract Semilinear Parabolic Equations 317~~

~~§ 4 Determining Functionals for Abstract Semilinear Parabolic Equations~~

In this section we prove a number of assertions on the existence and properties of determining functionals for processes generated in some separable Hilbert space H by an equation of the form u d + Au = Bu(), t,,t > 0 u = u . (4.1) dt t = 0 0 Here A is a positive operator with discrete spectrum (for definition see Section 2.1) and Bu(), t is a continuous mapping from DA()12 ´ R into H possessing the properties (), £ (), - (), £ ()r 12 ()- B 0 t M0 ,(4.2)Bu1 t Bu2 t M A u1 u2 Î ()12 12 £ r r for all tu and for all j DA such that A uj , where is an arbitrary ()r positive number, M0 and M are positive numbers. Assume that problem (4.1) is uniquely solvable in the class of functions W = C ()[0+, ¥); H Ç C ()[0+, ¥); DA()12 and is pointwise dissipative, i.e. there exists R > 0 such that 12 () £ ³ () A ut R whentt0 u (4.3) for all ut()Î W . Examples of problems of the type (4.1) with the properties listed above can be found in Chapter 2, for example. The results obtained in Sections 1 and 2 enable us to establish the following assertion.

Theorem 4.1. L = {}= ,,¼ , For the set of linear functionals lj : j 12 N on the space = ()12 = 12 (), ; W VDA with the norm . V A . H to be VV -asymptotically determining for problem (4.1) under conditions (4.2) and (4.3), it is sufficient that the completeness defect eL ()VH, satisfies the inequality -1 eL º eL ()VH, < MR() , (4.4) where M()r and R are the same as in (4.2) and (4.3).

Proof. () () W We consider two solutions u1 t and u2 t to problem (4.1) that lie in . By virtue of dissipativity property (4.3) we can suppose that 12 () £ ³ = , A uj t R ,,t 0 j 12 . (4.5) 318 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C ()= ()- () () h Let ut u1 t u2 t . If we consider ut as a solution to the linear problem a u p d + Au = ft()º Bu()()t , t - Bu()()t , t , t dt 1 2 e r then it is easy to find that 5 t t 1 1 ut()2 + ()tAu()t , u()t d £ us()2 + MR() A12u()t ×tu()t d 2 ò 2 ò s s for all ts³³0 . We use (2.11) to obtain that

12 12 2 12 2 2 MR()××eA uu£ L ××MR() A u ++d A u CR(),,L d []Nu() for any d > 0 , where ()= {}() = ,,¼ , Nu max lj u : j 12 N . Therefore, t ()2 + ()-ed - () 12 ()t 2 t £ ut 21 L MR ò A u d s t £ us()2 + CR(),,L d ò []Nu()()t 2 dt .(4.6) s

Using (4.4) we can choose the parameter d > 0 such that 1-ed - L MR()> 0 . Thus, we can apply Theorem 1.1 and find that under condition (4.4) equation t +1 []()()t - ()t 2 t = limNu1 u2 d 0 t ® ¥ ò t implies the equality ()- () = limu1 t u2 t 0 .(4.7) t ® ¥ In order to complete the proof of the theorem we should obtain 12 ()()- () = limA u1 t u2 t 0 (4.8) t ® ¥ from (4.7). To prove (4.8) it should be first noted that q ()()- () = limA u1 t u2 t 0 (4.9) t ® ¥ for any 0 £ q < 12¤ . Indeed, the interpolation inequality (see Exercise 2.1.12) - q q Aquu£ 12× A12 u 2 ,,0 £ q < 12¤ and dissipativity property (4.5) enable us to obtain (4.9) from equation (4.7). Now we use the integral representation of a weak solution (see (2.2.3)) and the method applied in the proof of Lemma 2.4.1 to show (do it yourself) that Determining Functionals for Abstract Semilinear Parabolic Equations 319

~~Abu ()t £ CR(), M ,,.1 <~~

~~Exercise 4.1 Show that if the hypotheses of Theorem 4.1 hold, then equation (4.9) is valid for all 0 £ q < 1 .~~

~~The reasonings in the proof of Theorem 4.1 lead us to the following assertion.~~

Corollary 4.1. Assume that the hypotheses of Theorem 4.1 hold. Then for any two weak ()12 () () (in DA ) solutions u1 t and u2 t to problem (4.1) that are bounded on the whole axis, ìü sup íýA12 u ()t : -¥ <

~~Proof. In the situation considered equation (4.6) implies that t ()2 + b 12 ()t 2 t £ ()2 ut L ò A u d us s~~

~~for all ts>b and some L > 0 . It follows that -b ()- ut() £ e L tsus(),.ts³ Therefore, if we tend s ® -¥ , then using (4.10) we obtain that ut() = 0 Î ()º () for all t R , i.e. u1 t u2 t .~~

It should be noted that Corollary 4.1 means that solutions to problem (4.1) that are bounded on the whole axis are uniquely determined by their values on the functionals {} lj . It was this property of the functionals lj which was used by Ladyzhenskaya [2] to define the notion of determining modes for the two-dimensional Navier-Stokes system. We also note that a more general variant of Theorem 4.1 can be found in [3].

Exercise 4.2 Assume that problem (4.1) is autonomous, i.e. Bu()º, t º () A (), Bu. Let be a global attractor of the dynamical system VSt generated by weak (in VDA= ()12 ) solutions to problem (4.1) and L = {}= ,,¼ assume that a set of functionals lj : j 1 N possesses 320 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C () () h property (4.4). Then for any pair of trajectories u1 t and u2 t a A ()() = ()() lying in the attractor the condition lj u1 t lj u2 t implies p ()º () Î = ,,¼ , t that u1 t u2 t for all t R and j 12 N . e r Theorems 4.1 and 2.4 enable us to obtain conditions on the existence of N deter- 5 mining functionals.

Corollary 4.2. Assume that the Kolmogorov N -width of the embedding of the space = ()12 (); < ()-1 VDA into H possesses the property N VH MR . Then there exists a set of asymptotically ()VV, ; W -determining functionals for problem (4.1) consisting of N elements.

Theorem 2.4, Corollary 2.1, and Exercise 2.6 imply that if the hypotheses of Theorem 4.1 hold, then the family of functionals L given by equation (2.13) is a ()VV, ; W - l > ()2 () determining set for problem (4.1), provided N +1 MR . Here MR and R are the constants from (4.2) and (4.3). It should be noted that the set L of the form (2.13) for problem (4.1) is often called a set of determining modes. Thus, Theo- rem 4.1 and Exercise 2.6 imply that semilinear parabolic equation (4.1) possesses a finite number of determining modes. When condition (4.2) holds uniformly with respect to r , we can omit the requirement of dissipativity (4.3) in Theorem 4.1. Then the following assertion is valid.

Theorem 4.2. Assume that a continuous mapping Bu(), t from DA()12 ´ R into H possesses the properties (), £ (), - (), £ 12 ()- B 0 t M0 , Bu1 t Bu2 t MA u1 u2 (4.11) Î ()12 L = {}= ,,¼ , for all uj DA . Then a set of linear functionals lj : j 12 N onVDA= ()12 is asymptotically ()VV, ; W -determining for problem (4.1), -1 provided eL = eL ()VH, < M .

Proof. -1 If we reason as in the proof of Theorem 4.1, we obtain that if eL < M , then () () for an arbitrary pair of solutions u1 t and u2 t emanating from the points u1 and u2 at a moment s the equation (see (4.6)) t t ut()2 + b ò A12 u()t 2 dt £ us()2 + CNuò []()()t 2 dt (4.12) s s ()=bbe()- () = (), ()L , is valid. Here ut u1 t u2 t ,L M and C M are positive constants, and ()= {}() = ,,¼ , Nu max lj u : j 12 N . Determining Functionals for Abstract Semilinear Parabolic Equations 321

Therefore, using Theorem 1.1 we conclude that the condition t +1 lim[]Nu()()t 2 dt = 0 (4.13) t ® ¥ ò t implies that t +1 ìü limíýut()2 + A12 u()t 2 dt = 0 .(4.14) t ® ¥ îþò t ()= ()- () Since ut u1 t u2 t is a solution to the linear equation u d + Au = ft() º Bu()()t , t - Bu()()t , t ,(4.15) dt 1 2 it is easy to verify that t A12 ut()2 £ A12 us()2 + 1 f ()t 2 dt (4.16) 2 ò s for ts³ . It should be noted that equation (4.16) can be obtained with the help \ of formal multiplication of (4.15) by u()t with subsequent integration. This conver- sion can be grounded using the Galerkin approximations. If we integrate equation (4.16) with respect to st from -1 to t , then it is easy to see that t t A12 ut() £ A12 us()2 ds + 1 f ()t 2 dt . ò 2 ò t -1 t -1 Using the structure of the function ft() and inequality (4.11), we obtain that t æöM2 A12 ut() £ 1 + A12 u()t 2 dt . èø2 ò t -1 Consequently, (4.14) gives us that 12 ()()- () = limA u1 t u2 t 0 . t ® ¥ Therefore, Theorem 4.2 is proved.

Further considerations in this section are related to the problems possessing inertial manifolds (see Chapter 3). In order to cover a wider class of problems, it is convenient to introduce the notion of a process. Let HSt be a real reflexive Banach space. A two-parameter family { ();, t t ³}t ; t, t Î R of continuous mappings acting in H is said to be evolutionary, if the following conditions hold: (a)Sts(), × Ss(), t = St(), t ,ts³³t ,St(), t = I . (), (b)St su0 is a strongly continuous function of the variable t . 322 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C (), (), t (), t h A pair HSt with St being an evolutionary family in H is often called a a process. Therewith the space H is said to be a phase space and the family p t of mappings St(), t is called an evolutionary operator. A curve e g []= {}()º ()+ , ³ r s u0 ut St s su0 : t 0

5 is said to be a trajectory of the process emanating from the point u0 at the moment s . (), It is evident that every dynamical system HSt is a process. However, main examples of processes are given by evolutionary equations of the form (1.1). There- with the evolutionary operator is defined by the obvious formula (), = (), ; St su0 ut s u0 , ()= (), ; where ut ut s u0 is the solution to problem (1.1) with the initial condition u0 at the moment s .

~~Exercise 4.3 Assume that the conditions of Section 2.2 and the hypotheses of Theorem 2.2.3 hold for problem (4.1). Show that weak solutions to problem (4.1) generate a process in H .~~

~~, Î Î Exercise 4.4 Show that for any two elements u1 u2 Ms , s R the following inequality holds - ()1 + L2 12 u - u ££Pu()- u u - u . 1 2 V 1 2 V 1 2 V~~

Exercise 4.5 Using equation (4.19) prove that ()1 - P St(), su - L()PS() t, s u , t £ ()1 + L × Ce-g ()ts- 0 0 V for ts³ . Determining Functionals for Abstract Semilinear Parabolic Equations 323

()= (), = , Exercise 4.6 Show that for any two trajectories uj t St s uj , j 12 , of the process ()VSt; (), t the condition ()()- () = limPu1 t u2 t 0 t ® ¥ V implies that ()- () = limu1 t u2 t 0 . t ® ¥ V

In particular, the results of these exercises mean that Ms is homeomorphic to a subset in RN , NP= dim , for every s Î R . The corresponding homeomorphism r : V ® RN can be defined by the equality ru=f{}() u, f N , where {} is a ba- j V j = 1 j sis in PV . Therewith the set of functionals {} l ()u ==()u, j : j 1,,¼ N appears j j V to be asymptotically determining for the process. The following theorem contains {} a sufficient condition of the fact that a set of functionals lj possesses the properties mentioned above.

Theorem 4.3. Assume that VH and are separable Hilbert spaces such that V is continuously and densely embedded into HVSt. Let a process (); (), t possess {} an asymptotically complete finite-dimensional inertial manifold Mt . {} Assume that the orthoprojector PM from the definition of t can be continuously extended to the mapping from H into V, i.e. there exists a constant LL= ()P > 0 such that £ L × Î Pv V v H ,,.vV. (4.20) L = {}= ,,¼ If lj : j 1 N is a set of linear functionals on V such that 2 -12 -1 eL ()VH; < ()1+ L L , (4.21) then the following conditions are valid: L 1) there exist positive constants c1 and c2 depending on such that c u - u ££max {}l ()u - u : j = 1,,¼ N c u - u (4.22) 1 1 2 H j 1 2 2 1 2 H , Î Î for all u1 u2 Mt , t R ; i.e. the mapping rL acting from V into N according to the formula r ul= {}()u N is a Lipschitzian ho- R L j j = 1 N Î meomorphism from Mt into R for every t R ; 2) the set of functionals L is determining for the process ()VSt; (), t in ()= (), the sense that for any two trajectories uj t St suj the condition ()()() - ()() = limlj u1 t lj u2 t 0 t ® ¥ implies that ()- () = limu1 t u2 t 0 . t ® ¥ V 324 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C h Proof. a Let u , u Î M . Then p 1 2 t t u = Pu + L()Pu , t ,.j = 12, e j j j r Therewith equation (4.17) gives us that 5 u - u £ ()1 + L2 12 Pu - Pu ,.(4.23)u Î M 1 2 V 1 2 V j t Consequently, using Theorem 2.1 and inequality (4.20) we obtain that u - u £ C N ()eu - u + ()1 + L2 12× L u - u , 1 2 H L L 1 2 L 1 2 H ()= {}() = ,,¼ e = e (); where NL u max lj u : j 1 N and L L VH . Therefore, equation (4.21) implies that u - u £ C ()L × N ()u - u , 1 2 H 1 L 1 2 (4.24) ()L = × ()- e ()+ 2 12 L -1 whereC1 CL 1 L 1 L . On the other hand, (4.20) and (4.23) give us that l ()u - u ££C u - u C ()1+ L2 12L u - u .(4.25) j 1 2 L 1 2 V L 1 2 H Equations (4.24) and (4.25) imply estimate (4.22). Hence, assertion 1 of the theorem is proved. ()= (), ³ Let us prove the second assertion of the theorem. Let uj t Stsuj , ts , be trajectories of the process. Since ()= ()()+ L()(), + ()()- ()- L()(), uj t Puj t Puj t t 1 P uj t Puj t t , using (4.17) it is easy to find that u ()t - u ()t £ ()1 + L2 12 Pu()()t - u ()t + 1 2 V 1 2 V

+ ()1 - P u ()t - L()Pu ()t , t . å j j V j = 12, The property of asymptotical completeness (4.19) implies (see Exercise 4.5) that ()- ()- L()(), £ -g ()ts- ³ 1 P uj t Puj t t Ce ,.ts Therefore, equation (4.20) gives us the estimate -g ()- u ()t - u ()t £ ()1 + L2 12L u ()t - u ()t + Ce ts.(4.26) 1 2 V 1 2 H It follows from Theorem 2.1 that u ()t - u ()t £ C N ()eu ()t - u ()t + u ()t - u ()t . 1 2 H L L 1 2 L 1 2 V Therefore, provided (4.21) holds, equation (4.26) implies that -g ()- u ()t - u ()t £ A × N ()u ()t - u ()t + B e ts,,ts³ 1 2 V L L 1 2 L Determining Functionals for Abstract Semilinear Parabolic Equations 325

~~where AL and BL are positive numbers. Hence, the condition ()()- () = limNL u1 t u2 t 0 t ® ¥ implies that ()- () = limu1 t u2 t 0 . t ® ¥ V Thus, Theorem 4.3 is proved.~~

~~Exercise 4.7 Assume that the hypotheses of Theorem 4.3 hold. Let , Î ()= () ÎL u1 u2 Ms be such that lj u1 lj u2 for all lj . Show that (), º (), ³ St su1 St su2 forts .~~

Exercise 4.8 Prove that if the hypotheses of Theorem 4.3 hold, then inequality (4.22) as well as the equation c u - u ££max {}l ()u - u : j = 12,,¼ ,N c u - u 1 1 2 V j 1 2 2 1 2 V , Î Î , > is valid for any u1 u2 Mt and t R , where c1 c2 0 are constants depending on L .

Let us return to problem (4.1). Assume that Bu(), t is a continuous mapping from DA()q ´ R into H , 0 £ q < 1 , possessing the properties (), £ æö+ q (), - (), £ q()- Bu0 t M èø1 A u0 , Bu1 t Bu2 t MA u1 u2 Î ()q £ q < for all uj DA , 0 1 . Assume that the spectral gap condition 2M q q l - l ³ ()()l1 + k + l n +1 n q n +1 n <<- {}l holds for some n and 0 q 22 . Here n are the eigenvalues of the operator Ak indexed in the increasing order and is a constant defined by (3.1.7). Under these conditions there exists (see Chapter 2) a process ()DA()q ; St(), s generated by problem (4.1). By virtue of Theorems 3.2.1 and 3.3.1 this process possesses an {} asymptotically complete finite-dimensional inertial manifold Mt and the corresponding orthoprojector Pn is a projector onto the span of the first eigenvectors of the operator A . Therefore, q £ lq -s s -¥ <

C h Due to Theorem 2.4 this estimate can be rewritten as follows: a q - 1- q l + s p e ()()q ; ()s £ ××æön 1 ()()q ; ()s L DA DA n DA DA , t 2 èøl e 22- q + 2 q n r ()()q ; ()s ()q where n DA DA is the Kolmogorov nDA -width of the embedding of 5 into DA()s , - ¥< ~~ut uxxxx uxx ux u 0 ,,,x 0 L t 0 [], {} with the periodic boundary conditions on 0 L in the case when lj is a set of uniformly distributed nodes on []0, L , i.e.~~

~~~~l ()u = ujh(),whereh = L ,j = 01,,¼, N -1 . j N For the references and discussion of general case see survey [3].~~~~

In conclusion of this section we give one more theorem on the existence of determining functionals for problem (4.1). The theorem shows that in some cases we can require that the values of functionals on the difference of two solutions tend to zero only on a sequence of moments of time (cf. Theorem 1.3).

~~~~Theorem 4.4. As before, assume that A is a positive operator with discrete spectrum: = l~~~~

~~~~Proof. ()= ()- () Let ut u1 t u2 t . Then the results of Chapter 2 (see Theorem 2.2.3 and Exercise 2.2.7) imply that a ts- q () £ 2 q () A ut a1 e A us ,(4.28)~~~~

ìü-l ts- a a ts- ()1- P Aq ut() £ íýe n +1 + 3 e 2 Aq us() (4.29) n îþl1-q n +1 ³³ ql, for ts0 , where a1 , a2 , and a3 are positive numbers depending on 1 , and {},,¼ MP and n is the orthoprojector onto Lin e1 en . It follows form (4.29) that ()- q () £ q () +a ££+ b 1 Pn A ut qab, A us ,,(4.30)s ts where -al a b n +1 3 a2 qab, = e + e . l1 - q n +1 = ()abql,,, , £ ¤ Let us choose nn 1 M such that qab, 12 . Then equation (4.30) gives us that q Aq ut()£ l ut() + 1 Aqus(),.s + a ££ts+ b n 2 This inequality as well as estimate (4.28) enables us to use Theorem 1.3 with V = = DA()q and to complete the proof of Theorem 4.4.

~~~~Exercise 4.9 Assume that nq is chosen such that ab, < 1 in the proof ()()- ()® of Theorem 4.4. Show that the condition Pn u1 tk u2 tk 0 as k ® ¥ implies (4.27).~~~~

The results presented in this section can also be proved for semilinear retarded equations. For example, we can consider a retarded perturbation of problem (4.1) of the following form ì du + = (), + (), ï Au Bu t Qut t , í dt ï u = j()t Î C º Cr()[]- , 0 , DA()12¤ , î trÎ []- , 0 r where, as usual (see Section 2.8), ut is an element of Cr defined with the help () ()q =q()+ q Î []- , ofut by the equality ut ut , r 0 , and Q is a continuous map- ´ ping from Cr R into H possessing the property 0 (), - (), 2 £ × 12 ()q - ()q 2 q Qv1 t Qv2 t M1 ò A v1 v2 d -r , Î for any v1 v2 Cr . The corresponding scheme of reasoning is similar to the method used in [3], where the second order in time retarded equations are considered. 328 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C h § 5 Determining Functionals a p for Reaction-Diffusion Systems t e r 5 In this section we consider systems of parabolic equations of the reaction-diffusion type and find conditions under which a finite set of linear functionals given on the phase space uniquely determines the asymptotic behaviour of solutions. In particular, the results obtained enable us to prove the existence of finite collections of determining modes, nodes, and local volume averages for the class of systems under consideration. It also appears that in some cases determining functionals can be given only on a part of components of the state vector. As an example, we consider a system of equations which describes the Belousov-Zhabotinsky reaction and the Navier-Stokes equations. Assume that W is a smooth bounded domain in Rn , n ³ 1 , Hs()W is the W s ()W s ()W Sobolev space of the order s on , and H0 is the closure (in H ) of the set W . of infinitely differentiable functions with compact support in . Let s be a norm in Hs()W and let . and () . , . be a norm and an inner product in L2()W , respectively. Further we also use the spaces Hs = ()Hs ()W m º Hs ()W ´ ¼´ Hs ()W ,.m ³ 1 2 s Notations L and H0 have a similar meaning. We denote the norms and the inner products in L2 and Hs as in L2()W and Hs ()W . We consider the following system of equations ¶ = (), D - (),,Ñ ; Î W > tuaxtu fxu ut,,,(5.1)x t 0 = (), = () u ¶W 0 ,,ux 0 u0 x (), =D()()¼, ; ; (), as the main model. Here ux t u1 xt um xt , is the Laplace operator, Ñu = ()¶ u ,,¼¶ u , and axt(), is an mm -by- matrix with the ele- k x1 k xn k ¥()W ´ Î W Î ments from L R+ such that for all x and t R+ a ()xt, ºm1 ×m()aa+ * ³ × I ,.> 0 (5.2) + 2 0 0 We also assume that the continuous function = (); ¼ ; W ´ ()n +1 m ´ ® m ff1 fm : R R+ R is such that problem (5.1) has solutions which belong to a class W of functions onR+ with the following properties: Î W > a) for any u there exists t0 0 such that ()Î¶(), ¥ ; 2 Ç 1 ()Î (), ¥ ; 2 ut Ct0 + H H0 ,,(5.3)t ut Ct0 + L where Cab(), ; X is the space of strongly continuous functions on []ab, with the values in X ; Determining Functionals for Reaction-Diffusion Systems 329

~~~~b) there exists a constant k > 0 such that for any uv, Î W there exists > > t1 0 such that for tt1 we have (), Ñ ; - (), Ñ ; £ × æö-Ñ+ - Ñ fu ut fv vt kuvèøu v .(5.4)~~~~

It should be noted that if axt(), is a diagonal matrix with the elements from 2 ()W ´ C R+ and f is a continuously differentiable mapping such that fxu(),, p; t - fxvq(), , ; t £ kuv× ()- + pq- (5.5) , Î m , Î nm Î W Î for all uv R , pq R , x , and t R+ , then under natural compatibility conditions problem (5.1) has a unique classical solution [7] which evidently possesses properties (5.3) and (5.4). In cases when the dynamical system generated by equations (5.1) is dissipative, the global Lipschitz condition (5.5) can be weakened. For example (see [8]), if a is a constant matrix and n fxu(),,Ñut; = fxu(), + b ()¶x ugx+ (), å j xj j = 1 = ()1 ,,¼ m Î ¥ = (),,¼ Î 2 = (),,¼ where bj bj bj L ,gg1 gm L , and f f1 fm is continuously differentiable and satisfies the conditions p p - 1 (), ³ m 0 (), £ m 0 + > f xuu 1 u ,fx u 2 u C ,,p0 2

¶f p k £ C × ()1 + u 1 ,,,1 ££km 1 ££jn ¶ uj m , m > < ()¤ , ¤ ()- > where 1 2 0 and p1 min 4 n 2 n 2 for n 2 , then any solution to problem (5.1) with the initial condition from L2 is unique and possesses properties (5.3) and (5.4).

~~~~Let us formulate our main assertion.~~~~

Theorem 5.1. L = {}= ,,¼ A set lj : j 1 N of linearly independent continuous linear 2 Ç 1 functionals on H H0 is an asymptotically determining set with respect 1 W to the space H0 for problem (5.1) in the class if c m 2 -12 e ()2 Ç 1, 2 < 0 0 × æö+ 27 × æök º (), m L H H0 L èø1 èøm pk 0 , (5.6) k 2 4 0 -1 = {}Î ()W2 Ç 1 () D £ m where c0 sup w 2 : w H H0 , w 1 , and 0 and k are constants from (5.2) and (5.4). This means that if inequality (5.6) holds, then for some pair of solutions uv, Î W the equation t +1 ()()t - ()()t 2 t = = ,,¼ limlj u lj v d 0 ,,,j 1 N , (5.7) t ® ¥ ò t 330 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C 1 h implies their asymptotic closeness in the space H : a ()- () = limut vt 1 0 . (5.8) p t ® ¥ t e r Proof. 5 Let uv, Î W . Then equation (5.1) for wuv= - gives us that ¶ = (), D - ()(),,Ñ ; - (),,Ñ ; t waxtw fxu ut fxv vt. If we multiply this by Dw in L2 scalarwise and use equation (5.4), then we find that æö 1 ×md Dwt()2 + Dwt()2 £ kwt() + ()Ñw ()t × Dwt() 2 dt 0 èø for t >Ñ0 large enough. Therefore, the inequality w 2 = ()ww, D £ ww× D enables us to obtain the estimate 2 2 d Ñ 2 + m D 2 £ 2 k æö+ 27 æök 2 w 0 w m èø1 èøm w .(5.9) dt 0 4 0 Theorem 2.1 implies that 2 2 2 2 w £ CN(), d max l ()w + ()e1+ d ××L w (5.10) j j 2 Îd2 Ç 1 > (), d >eº for all w H H0 and for any 0 , where CN 0 is a constant and L º e ()2 Ç 1, 2 L H H0 L . Consequently, estimate (5.9) gives us that ()e+ d 2 d 2 æö1 L 2 2 Ñwt() + m ××ç÷1 - Dwt() £ Cl× max ()w , dt 0 (), m 2 j j èøpk 0 (), m where pk 0 is defined by equation (5.6). It follows that if estimate (5.6) is valid, then there exists b > 0 such that t -b ()tt- Ñ ()2 £ 0 Ñ ()2 + × -b ()t -t ()()t 2 t wt e wt0 Ce max lj w d ò j t0 ³ for all tt0 , where t0 is large enough. Therefore, equation (5.7) implies (5.8). Thus, Theorem 5.1 is proved.

Exercise 5.1 Assume that ut() and vt() are two solutions to equation (5.1) Î Î defined for all t R . Let (5.3) and (5.4) hold for every t0 R and let æö sup Ñut()+ Ñvt() < ¥ . t < 0 èø Prove that if the hypotheses of Theorem 5.1 hold, then equalities ()() = ()() = ,,¼ < £ ¥ lj ut lj vt for j 1 N and ts for some s imply that ut()º vt() for all ts< . Determining Functionals for Reaction-Diffusion Systems 331

~~~~Let us give several examples of sets of determining functionals for problem (5.1).~~~~

Example 5.1 (determining modes, m ³ 1 ) {}- D 2 Let ek be eigenelements of the operator in L with the Dirichlet bounda- ¶W < l ££l ¼ ry conditions on and let 0 1 2 be the corresponding eigenvalues. Then the completeness defect of the set ìü L = íýl : l ()w = wx()× e ()x dx , j = 1,,¼ N îþj j ò j W e ()2 Ç 1, 2 £ × l-1 can be easily estimated as follows: L H H0 L n N +1 (see Exer- l > × (), m -1 L cise 2.6). Thus, if N possesses the property N +1 npk 0 , then ()2 Ç 1,,1 W is a set of asymptotically H H0 H0 -determining functionals for problem (5.1).

~~~~Considerations of Section 5.3 also enable us to give the following examples.~~~~

Example 5.2 (determining generalized local volume averages) Assume that the domain W is divided into local Lipschitzian subdomains {}W = ,,¼ > j : j 1 N , with diameters not exceeding some given number h 0 . W l ()Î ¥ ()W Assume that on every domain j a function j x L j is given such W l that the domain j is star-like with respect to supp j and the conditions L l () = l () £ j x dx 1 ,,ess sup j x n ò x Î W h W j j hold, where the constant L > 0 does not depend on h and j . Theorem 3.1 implies that 2 1 2 2 2 eL ()H ()W Ç H ()W ; L ()W £ c h × L h 0 n for the set of functionals ìü L = íýl ()u = l()x ux()dx , j = 12,,¼ ,N . h îþj ò j W j = s2 -2 Therewith the reasonings in the proof of Theorem 3.1 imply that cn n n , s n ÎL where n is the area of the unit sphere in R . For every lj h we define the ()k 2 functionals lj on H by the formula ()k ()= () = (),,¼ Î 2 = ,,¼ , lj w lj wk ,,.(5.11)ww1 wm H k 12 m Let L ()m = {}()k () = ,,¼ , ÎL h lj u : k 12 m , lj h .(5.12) 332 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C h One can check (see Exercise 2.8) that a 1 p e ()()2 Ç ; 2 £ ××2 L2 . L m H H0 L cm h t h e L ()m r Therefore, if h is small enough, then h is a set of asymptotically deter- 5 mining functionals for problem (5.1).

Example 5.3 (determining nodes, n £ 3 ) Let W be a convex smooth domain in Rn , n £ 3 . Let h > 0 and let W = W Ç {}= (),,¼ <<()+ = ,, j xx1 xn : jk hxk jk 1 h , k 123, = (),,¼ Î n Ç W where j j1 jn Z . Let us choose a point xj in every subdomain W j and define the set of nodes L = {}()= () Î n Ç W £ h lj u uxj : j Z ,.n 3 (5.13) Theorem 3.2 enables us to state that 2 1 2 2 eL ()H ()W Ç H ()W ; L ()W £ ch× , h 0 L ()m where c is an absolute constant. Therefore, the set of functionals h defined L by formulae (5.11) and (5.12) with h given by equality (5.13) possesses the property 1 e ()()2 Ç ; 2 £ × 2 L m H H0 L ch. h L ()m Consequently, h is a set of asymptotically determining functionals for problem (5.1) in the class W , provided that h is small enough.

It is also clear that the result of Exercise 2.8 enables us to construct mixed determining functionals: they are determining nodes or local volume averages depending on the components of a state vector. Other variants are also possible.

However, it is possible that not all the components of the solution vector ux(), t appear to be essential for the asymptotic behaviour to be uniquely determined. A theorem below shows when this situation can occur. Let I be a subset of {}1,,¼ m . Let us introduce the spaces s = {}= (),,¼ Î s º , Ï Î HI ww1 wm H : wk 0 kI,.s R We identify these spaces with ()Hs ()W I , where I is the number of elements of 2 s L the set I . Notations LI and H0, I have the similar meaning. The set of linear 2 *L functionals on HI is said to be determining if pI is determining, where pI is the s s natural projection of H onto HI . Determining Functionals for Reaction-Diffusion Systems 333

Theorem 5.2. = (),,¼ Let a diag d1 dm be a diagonal matrix with constant elements and let {}II, ¢ be a partition of the set {}1,,¼ m into two disjoint subsets. w,,* q = ,,¼ Assume that there exist positive constants k i , where i 1 m , such that for any pair of solutions uv, Î W the following inequality holds (hereinafter wuv= - )):: ìüd q - i D 2 + ()(), Ñ ; - (), Ñ ; , D + å i íýwi fi u ut fi v vt wi îþ2 iIÎ ìü + q - Ñ 2 - ()(), Ñ ; - (), Ñ ; , £ å i íýdi wi fi u ut fi v vt wi îþ iIÎ ¢

~~~~£ -w 2 + * 2 å wi k å wi . (5.14) iIÎ ¢ iIÎ~~~~

Then a set = {lj : j = 340() 1 ,, ¼ N of linearly independent continuous linear 2 Ç 1 functionals on HI H0, I is an asymptotically determining set with respect 1 W to the space H0 for problem (5.1) in the class if d q e º e ()2 Ç 1 , 2 < × i i L L HI H0, I LI c0 min , (5.15) iIÎ 2 k* > where c0 0 is defined as in (5.6). This means that if two solutions uv, Î W possess the property t +1 ()()t - ()()t 2 t = = ,,¼ limlj pI u lj pI v d 0 for j 1 N , (5.16) t ® ¥ ò t s s where pI is the natural projection of H onto HI , then equation (5.8) holds.

Proof. Let uv, ÎW andwuv= - . Then ¶ = D - ()(),,Ñ ; - (),,Ñ ; t wi di wi fi xu ut fi xv vt (5.17) = ,,¼ 2()W - qD for i 1 m . In L we scalarwise multiply equations (5.17) by i wi Îq Î ¢ foriI and by i wi for iI and summarize the results. Using inequality (5.14) we find that

1 × d L()wt() ++1 d q Dw 2 w w 2 £ k* w 2 , 2 dt 2 å i i i å i å i iIÎ iIÎ ¢ iIÎ where L()= qÑ 2 + q 2 w å i wi å i wi . iIÎ iIÎ ¢ 334 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C h As in the proof of Theorem 5.1 (see (5.10)) we have a + d p 2 £ (), d ()2 + 1 e2 × D 2 å wi CN max lj pI w L å wi t j c2 e iIÎ 0 iIÎ r for every d > 0 . Therefore, provided (5.15) holds, we obtain that 5 d L()bL() + ()() £ × ()() 2 wt wt Clmax j pI wt dt j with some constant b >L0 . Equation (5.16) implies that ()wt() ® 0 as t ® 0 . Hence, limwt() = 0 .(5.18) t ® ¥ However, equation (5.9) and the inequality Ñ £ × D Î 2 Ç 1 wCw,,w H H0 imply that t -b ()tt- -b ()tt- Ñ ()2 £ 0 Ñ ()2 + 0 ()t 2 t wt e wt0 Ceò w d

~~~~t0 ³ b > for all tt0 with t0 large enough and for 0 . Therefore, equation (5.8) follows from (5.18). Theorem 5.2 is proved.~~~~

~~~~The abstract form of Theorem 5.2 can be found in [3].~~~~

As an application of Theorem 5.2 we consider a system of equations which describe the Belousov-Zhabotinsky reaction. This system (see [9], [10], and the references there- £ = (), = (),, in) can be obtained from (5.1) if we take n 3 , m 3 , axt diag d1 d2 d3 and (),,Ñ ; º ()= ()(); (); () fxu ut fu f1 u f2 u f3 u , where ()= - a()- + - b 2 f1 u u2 u1u2 u1 u1 , ()= - 1 ()g - - ()= - d()- f2 u a u3 u2 u1u2 ,.f3 u u1 u3 Here abg,,, and d are positive numbers. The theorem on the existence of classical solutions can be proved without any difficulty (see, e.g., [7]). It is well-known >> (), b-1 > g [10] that if a3 a1 max 1 and a2 a3 , then the domain = {}º (),, ££ ,, Ì 3 Duu1 u2 u3 : 0 uj aj , j =1 2 3 R is invariant (if the initial condition vector lies in Dx for all Î W , then uxt(), Î D ÎW > WWº for x and t 0 ). Let D be a set of classical solutions the initial conditions of which have the values in D . It is clear that assumptions (5.3) and (5.4) are W w,,* q q valid for . Simple calculations show that the numbers k 2 , and 3 can be Determining Functionals for Reaction-Diffusion Systems 335

= {} ¢ =q{}, = chosen such that equation (5.14) holds for I 1 , I 23 , and 1 1 . In- deed, let smooth functions ux() and vx() be such that ux(), vx()Î D for all x Î W and let wx()= ux()- vx() . Then it is evident that there exist constants > Cj 0 such that d æö L º ()f ()u - f ()v , Dw £ 1 Dw 2 + C × w 2 + w 2 , 1 1 1 1 2 1 1 èø1 2 æö L º -()f ()u - f ()v , w £ -1 w 2 + C × w 2 + w 2 , 2 2 2 2 2a 2 2 èø1 3 d L º -()f ()u - f ()v , w £ - w 2 + C × w 2 . 3 3 3 3 2 3 3 1 q , q > Consequently, for any 2 3 0 we have d L ++q L q L £ 1 Dw 2 ++()C ++q C q C w 2 1 2 2 3 3 2 1 1 2 2 3 3 1 æöq æöq d ++C - 2 w 2 q C - 3 w 2 . èø1 2a 2 èø2 2 2 3 q q It follows that there is a possibility to choose the parameters 2 and 3 such that d æö L ++q L q L £ 1 Dw 2 + k* w 2 - w w 2 + w 2 1 2 2 3 3 2 1 1 èø2 3 with positive constants k* and w . This enables us to prove (5.14) and, hence, the validity of the assertions of Theorem 5.2 for the system of Belousov-Zhabotinsky L = {}= ,,¼ equations. Therefore, if lj : j 1 N is a set of linear functionals on 2 ()W Çe1()W ()()W2 Ç 1 (), 2()W H H0 such that L H H0 L is small enough, then the condition t +1 ()()t - ()()t 2 t = = ,,¼ limlj u1 lj v1 d 0 ,,j 1 N t ® ¥ ò t ()= ()(),,() () ()= (,(), () for some pair of solutions ut u1 t u2 t u3 t and vt v1 t v2 t ()) W v3 t which lie in implies that ()- () ® ® ¥ ut vt 1 0 ast . In particular, this means that the asymptotic behaviour of solutions to the Belousov- Zhabotinsky system is uniquely determined by the behaviour of one of the components of the state vector. A similar effect for the other equations is discussed in the sections to follow.

The approach presented above can also be used in the study of the Navier-Stokes system. As an example, let us consider equations that describe the dynamics of a vis- cous incompressible fluid in the domain W º T 2 = ()0, L ´ ()0, L with periodic boundary conditions: 336 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C ¶ - nD ++(), Ñ Ñ = (), Î 2 > h t u u u u p Fxt,,,xT t 0 a p Ñ = Î 2 (), = () t u 0, xT, ux 0 u0 x , (5.19) e (), = ()(), ; (), r where the unknown velocity vector ux t u1 xt u2 xt and pressure (), n > 5 px t are L -periodic functions with respect to spatial variables, 0 , and Fx(), t is the external force. Let us introduce some definitions. Let V be a space of trigonometric polynomials vx() of the period L with the values in R2 such that div v = 0 and ò vx()dx = 0 . Let H be the closure of V in L2 , let P be the orthoprojector onto T2 H in L2 , let A ==-P Du -Du and Bu()P, v = ()u, Ñ v for all uv and from DA()= H Ç H2 . We remind (see, e.g., [11]) that A is a positive operator with discrete spectrum and the bilinear operator Bu(), v is a continuous mapping from DA()´ DA() into H . In this case problem (5.19) can be rewritten in the form ¶ u ++nAu B() u, u = PFt(),.(5.20)u = u ÎH t t = 0 0 ÎP()Î ¥(); It is well-known (see, e.g., [11]) that if u0 H and Ft L R+ H , then problem (5.20) has a unique solution ut() such that ()Î (); Ç (), ¥ ; () > ut C R+ H Ct0 + DA ,.(5.21)t0 0 One can prove (see [9] and [12]) that it possesses the property ta+ 2 æö lim 1 Au()t 2 dt £ F 1 + 1 (5.22) a ò n2 èønl t ® ¥ a 1 t >l= ()p¤ 2 for any a 0 . Here 1 2 L is the first eigenvalue of the operator A in H and F = lim PFt() . t ® ¥

~~~~Lemma 5.1. Let uv, Î DA() and let wuv= - . Then~~~~

()Bu(), u - Bv(), v, Aw £ 2 w ¥ ÑwAu, (5.23) ()(), - (), , £ Ñ ×××Ñ 12 D 12 Bu u Bv v Aw cL wk w w Au , (5.24) . ¥ where ¥ is the L -norm, wk is the kw -th component of the vector , k = 12, , and c is an absolute constant.

Proof. Using the identity (see [12]) ()Bu(), w, Aw ++()Bw(), v, Aw ()Bw(), w, Au = 0 for uv, Î DA() and wuv= - , it is easy to find that ()Bu(), u - Bv(), v, Aw = -()Bw(), w, Au . Determining Functionals for Reaction-Diffusion Systems 337

Therefore, it is sufficient to estimate the norm Bw(), w . The incompressibili- ty condition Ñw = 0 gives us that (), Ñ = ()- ¶ + ¶ ¶ - ¶ w ww1 2 w2 w2 2 w1 , w1 1 w2 w2 1 w1 , ¶ where i is the derivative with respect to the variable xi . Consequently, (), Ñ 2 £ æö(), Ñ 2 + (), Ñ 2 w w 2 èøw1 w2 w2 w1 . (5.25) This implies (5.23). Let us prove (5.24). For the sake of definiteness we let k = 1 . We can also assume that w Î V . Then (5.25) gives us that æö ()w, Ñ w £ 2 w Ñw + w Ñw . èø1 L4 2 L4 2 ¥ 1 We use the inequalities (see, e.g., [11], [12]) v £ a v 12× Dv 12 L¥ 0 and v £ a v 12× Ñv 12, L4 1

~~~~where a0 and a1 are absolute constants (their explicit equations can be found in [12]). These inequalities as well as a simply verifiable estimate v £ £ ()ÑL¤ 2 p × v imply that~~~~

~~~~(), Ñ £ L ()Ñ2 + ××Ñ 12 D 12 w w p a1 a0 w1 w w . This proves (5.24) for k = 1 .~~~~

Theorem 5.3. L = {}= ,,¼ , 1. A set lj : j 12 N of linearly independent continuous linear functionals on DA()= H2 Ç H is an asymptotically determining set with respect to H1 for problem (5.20) in the class of solutions with property (5.21) if -1 e º e ()(), < º n2 æöP () L L DA H c1Gc1 èølim Ft , (5.26) t ® ¥ where c1 is an absolute constant. L = {}= ,,¼ 2()W 2. Let lj : j 1 N be a set of linear functionals on H and let -2 e¢ º e ()2 ()W , 2()W < n4 -3 æöP () L L H L c2 L èølim Ft , t ® ¥ * L where c2 is an absolute constant. Then every set pk is an asymptotically determining set with respect to H1 for problem (5.20) in the class of solutions with property (5.21). Here pk is the natural projection onto the k -th (); = = , component of the velocity vector, pk u1 u2 uk ,,. k 12. 338 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C h Proof. a Let ut() and vt() be solutions to problem (5.20) possessing property (5.21). p t Then equations (5.20) and (5.23) imply that e r 1 d 2 2 ×n Ñw + Dw £ 2 w ¥ ××ÑwAu 5 2 dt for wuv= - . As above, Theorem 2.1 gives us the estimate

wC£ h()w + eL Dw ,.h()w = max l ()w j j Therefore, ££××12 D 12 h()× D 12 + e × D w ¥ a0 w w c1 w w a0 L w . The inequality Ñw 2 £ ww× D 12 12 enables us to obtain the estimate (see Exercise 2.12) eL ()eDA(), DA()£ L and hence 2 2 -1 2 Dw ³ -hCd ()w + ()e1 - d L × Ñw for every d > 0 . Therefore, we use dissipativity properties of solutions (see [8] and [9] as well as Chapter 2) to obtain that

d Ñwt()2 + a()t × Ñw 2 £ C × h()w 2 + h()w , dt where a ()= n()e- d -1 - 2 e n-1 ()2 t 1 L 2 a0 L Au t . =a()nl -1 () Equation (5.22) for a 1 implies that the function t possesses properties (1.6) and (1.7), provided (5.26) holds. Therefore, we apply Lemma 1.1 to obtain that Ñwt() ® 0 as t ® ¥ , provided t +1 lim[]h()w()t 2 dt = 0 . t ® ¥ ò t In order to prove the second part of the theorem, we use similar arguments. For the sake of definiteness let us consider the case k = 1 only. It follows from (5.20) and (5.24) that 1 ×nd Ñw 2 + Dw 2 £ cLÑw 12× Ñw ××Dw 12 Au .(5.27) 2 dt 1 The definition of completeness defect implies that Ñ 12 ££h()+ e¢D 12 h ()+ e¢D 12 w1 C w1 L w1 C w1 L w , where h ()w = max l ()w .Therefore, 1 j j 1 Determining Functionals in the Problem of Nerve Impulse Transmission 339

~~~~Ñ 12 Ñ × D 12 £ cL w1 wwAu~~~~

££×××e¢ Ñ D × + h()Ñ××D 12 cL L wwAu C w1 ww Au æö2 n £ C []h()w 2 ++ + Lc e¢ ××Ñw 2 Au 2 Dw 2 1 èøn L 2 for any > 0 , where the constant C depends on n,,L , and L . Consequently, from (5.27) we obtain that

~~~~d 2 2 2 æöLc2 2 2 Ñw + n Dw £ C []h ()w + 2 + e¢L × Ñw Au . dt 1 èøn 2 -1 Therefore, we can choose d= ××Lc ×n ×e¢L and find that~~~~

d 2 æö2 p 2 Lc2 2 2 2 Ñw + n -Ñ21()+ d e¢L × Au w £ C []h ()w , dt èøL n 1 where d > 0 is an arbitrary number. Further arguments repeat those in the proof of the first assertion. Theorem 5.3 is proved.

It should be noted that assertion 1 of Theorem 5.3 and the results of Section 3 enable us to obtain estimates for the number of determining nodes and local volume averages that are close to optimal (see the references in the survey [3]). At the same time, although assertion 2 uses only one component of the velocity vector, in general it makes it necessary to consider a much greater number of determining functionals in comparison with assertion 1. It should also be noted that assertion 2 remains true if instead of wk we consider the projections of the velocity vector onto an arbitrary a priori chosen direction [3]. Furthermore, analogues of Theorems 1.3 and 4.4 can be proved for the Navier-Stokes system (5.19) (the corresponding variants of estimates (4.28) and (4.29) can be derived from the arguments in [2], [8], and [9]).

~~~~§ 6 Determining Functionals in the Problem of Nerve Impulse Transmission~~~~

We consider the following system of partial differential equations suggested by Hodgkin and Huxley for the description of the mechanism of nerve impulse transmission: ¶ - ¶2 + (),,, = Î (), > t ud0 x u gu v1 v2 v3 0 ,,,(6.1)x 0 L t 0 ¶ - ¶2 + ()× ()- ()= Î(), > = ,, tvj dj x vj kj u vj hj u 0 ,,,.(6.2)x 0 L t 0 j 123 340 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C > ³ h Here d0 0 , dj 0 , and a 3 4 p ()g,,, = -g()d - -g()d - - ()d - guv1 v2 v3 1 v1 v2 1 u 2 v3 2 u 3 3 u ,(6.3) t e where g >d0 and >>>d 0 d . We also assume that k ()u and h ()u are the r j 1 3 2 j j ()> <<() given continuously differentiable functions such that kj u 0 and 0 hj u 1 , 5 = ,, j 123. In this model uv describes the electric potential in the nerve and j is the density of a chemical matter and can vary between 01 and . Problem (6.1) and (6.2) has been studied by many authors (see, e.g., [9], [10], [13] and the references therein) for different boundary conditions. The results of numerical simulation given in [13] show that the asymptotic behaviour of solutions to this problem can be quite complicated. In this chapter we focus on the existence and the structure of determining functionals for problem (6.1) and (6.2). In particular, we prove that the

asymptotic behaviour of densities vj is uniquely determined by sets of functionals defined on the electric potential uu only. Thus, the component of the state vector ( ,, ,) uv1 v2 v3 is leading in some sense. We equip equations (6.1) and (6.2) with the initial data = () = () = ,, u = u0 x ,,,(6.4)vj vj 0 x j 123 t 0 t = 0 and with one of the following boundary conditions: == = = > u = u = dj vj dj vj 0 ,,(6.5a)t 0 x 0 xL x = 0 xL= ¶ ==¶ × ¶ =× ¶ => x u x u dj x vj dj x vj 0 ,,(6.5b)t 0 x = 0 xL= x = 0 xL= ()+ , - (), ==× ()()+ , - (), Î 1 > ux L t ux t dj vj xLt vj xt 0 ,,,(6.5c)x R t 0 = ,, (), where j 123 . Thus, we have no boundary conditions for the function vj xt = ,, when the corresponding diffusion coefficient dj is equal to zero for some j 123 . Let us now describe some properties of solutions to problem (6.1)–(6.5). First of all it should be noted (see, e.g., [10]) that the parallelepiped ìü DUuv= íýº (),,,v v : d ££u d ,0££v 1 , j = 123,, Ì R4 îþ1 2 3 2 1 j is a positively invariant set for problem (6.1)–(6.5). This means that if the initial data ()= ()(),,,() () () Î [], U0 x u0 x v10 x v20 x v30 x belongs to Dx for almost all 0 L , then ()º ()(), ,,,(), (), (), Î Ut ux t v1 xt v2 xt v3 xt D for x Î []0, L and for all t > 0 for which the solution to problem (6.1)–(6.5) exists. ºº[]4 []2 (), 4 Let H0 H L 0 L be the space consisting of vector-functions ()º (),,, Î 2(), Î 2 (), = ,, Ux uv1 v2 v3 , where uL0 L , vj L 0 L , j 123 . We equip it with the standard norm. Let Determining Functionals in the Problem of Nerve Impulse Transmission 341

ìü H ()D = íýUx()Î H : Ux()Î D for almost all x Î ()0, L . 0 îþ0 Depending upon the boundary conditions (6.5 a, b, or c) we use the following nota- = []4 = []4 tions H1 V1 and H2 V2 , where = 1 (), 1(), 1 (), V1 H0 0 L ,orH 0 L ,orHper 0 L (6.6) and = 2(), Ç 1 (), V2 H 0 L H0 0 L ,or ìü ()Î 2(), ¶ 2 (), íýux H 0 L : x u = 0 ,orHper 0 L , (6.7) îþx ==0, x L respectively. Hereinafter Hs ()0, L is the Sobolev space of the order s on ()0, L , 1 1 1(), H0 and Hper are subspaces in H 0 L corresponding to the boundary conditions (6.5a) and (6.5c). The norm in Hs ()0, L is defined by the equality L 2 ==¶s 2 +¶2 æös ()2 + ()2 = ,,¼ u s x u u ò èøx ux ux dx , s 12 0 We use notations . and () ., . for the norm and the inner product in H º º L2()0, L . Further we assume that C ()0, TX; is the space of strongly continuous functions on []0, T with the values in XL . The notation p()0, TX; has a similar meaning. > = ,, Î () Let dj 0 for all j 123 . Then for every vector U0 H0 D problem ()Î () (6.1)–(6.5) has a unique solution Ut H0 D defined for all t (see, e.g., [9], [10]). This solution lies in (), ; () Ç 2 (), ; C 0 T H0 D L 0 T H1 [], Î ()Ç for any segment 0 T and if U0 H0 D H1 , then ()Î (), ; ()Ç Ç 2 (), ; Ut C 0 T H0 D H1 L 0 T H2 .(6.8) ()Ç Therefore, we can define the evolutionary semigroup St in the space H0 D H1 by the formula = ()º ()(), ,,,(), (), (), St U0 Ut ux t v1 xt v2 xt v3 xt , where Ut() is a solution to problem (6.1)–(6.5) with the initial conditions º ()(),,,() () () U0 u0 x v10 x v20 x v30 x . ()()Ç ; The dynamical system H0 D H1 St has been studied by many authors. In particular, it has been proved that it possesses a finite-dimensional global attractor [9]. 342 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C === h If d1 d2 d3 0 , then the corresponding evolutionary semigroup can be a defined in the space V = ()V ´ []H1 ()0, L 3 Ç H ()D . In this case for any segment p 1 1 0 t []0, T we have e 3 r = ()Î (), ; Ç 2 (), ; ´ ()1(), St U0 Ut C 0 T V1 L 0 TV2 H 0 L ,(6.9) 5 Î if U0 V1 . This assertion can be easily obtained by using the general methods of Chapter 2.

~~~~The following assertion is the main result of this section.~~~~

Theorem 6.1. L = {}= ,,¼ Let lj : j 1 N be a finite set of continuous linear functio- , = , nals on the space Vs s 12 (see (6.6) and (6.7)). Assume that () d e 1 º e ()V , H £ 0 (6.10) L L 1 + d0 K1 or

()2 d0 eL º eL ()V , H £ , (6.11) 2 2 + 2 d0 K2 where 3 b ()+ j Aj Bj K = ()d - d × (6.12) 1 1 2 å k* j = 1 j and æö3 æöb ()A + B 2 12 K = 2 × ç÷()g + g 2 + 5 ()d - d 2 × ç÷j j j (6.13) 2 èø1 2 1 2 å èø2 k* j = 1 j b =bg =bg = g with 1 3 1 ,,, 2 1 , 3 4 2 , and ìüìü A = max íý¶ k ()u : d ££u d ,,,B = max íý¶ ()k h ()u : d ££u d , j îþu j 2 1 j îþu j j 2 1 ìü k* = min íýk ()u : d ££u d . j îþj 2 1 L Then is an asymptotically determining set with respect to the space H0 for problem (6.1)–(6.5) in the sense that for any two solutions ()= ()(), ,,,, , , Ut ux t v1 xt v2 xt v3 xt and *()= ()*(), ,,,* (), * (), * (), U t u xt v1 xt v2 xt v3 xt > = = ,, satisfying either (6.8) with dj 0 , or (6.9) with dj 0 , j 123, the condition Determining Functionals in the Problem of Nerve Impulse Transmission 343

~~~~t +1 ()()t - ()*()t 2 t = = ,,¼ limlj u lj u d 0 for j 1 N (6.14) t ® ¥ ò t implies that ìü3 ()- *()2 + ()- * ()2 = limíýut u t å vj t vj t 0 . (6.15) t ® ¥ îþ j = 1~~~~

Proof. Assume that (6.10) is valid. Let ()= ()(), ,,,(), (), (), Ut ux t v1 xt v2 xt v3 xt and *()= ()*(), ,,,* (), * (), * (), U t u xt v1 xt v2 xt v3 xt > = = ,, be solutions satisfying either (6.8) with dj 0 , or (6.9) with dj 0 , j 123 . It is clear that (), * º (),,, - ()*,,,* * * = GUU gu v1 v2 v3 gu v1 v2 v3 = ()g 3 ++g 4 g ()- * + (), * 1 v1 v2 2 v3 3 uu hU U , where ()g, * = -g()d3 - * 3 * ()- * - ()d4 - * 4 ()- * hUU 1 v1 v2 v1 v2 1 u 2 v3 v3 2 u . Since UU, * Î D , it is evident that 3 (), * £ - * hU U å aj vj vj , j = 1 = ()bd - d × =y- * = - * = ,, where aj 1 2 j . Let wuu and j vj vj , j 123 . It follows from (6.1) that ¶ - ¶2 + (), * = Î (), > t wd0 x w GU U 0 ,,.(6.16)x 0 L t 0 If we multiply (6.16) by wL in 2()0, L , then it is easy to find that 3 1 × d w 2 ++d ¶ w 2 g w 2 £ a y × w .(6.17) 2 dt 0 x 3 å j j j = 1 Equation (6.2) also implies that

1 × d y 2 ++d ¶ y 2 k* y 2 £ ()yA + B × w .(6.18) 2 dt j j x j j j j j j < = ,, Thus, for any 0 1 and j 0 , j 123 , we obtain that 344 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C 3 3 h æö 2 2 2 2 * 2 a 1 × d ç÷+ q y +++¶ g e q y £ w å j j d0 x w 3 w å kj j j p 2 dt èø t j = 1 j = 1 e 3 3 r ££- ()- e *q y 2 + []+ q ()+ y × å 1 kj j j å aj j Aj Bj j w 5 j = 1 j = 1 3 []a + q ()A + B 2 £ å j j j j × w 2 . (6.19) 41()q- e k* j = 1 j j Theorem 2.1 gives us that

2 £ × ()2 + ()e+ h []()1 2 × æö¶ 2 + 2 w CL , h max lj w 1 L x w w j èø for any h > 0 . Therefore, æö ¶ 2 ³ 1 - × 2 - × ()2 x w ç÷1 w CL , h max lj w .(6.20) ()1 2 j èø()e1+ h []L Consequently, it follows from (6.19) that æö3 1 × d ç÷w 2 + q y 2 ++d × w()L,,,eqh w 2 2 dt èøå j j 0 j = 1 3 + e *q y 2 £ × ()2 kj j j CL , h max lj w (6.21) å j j = 1 <h> for any 0 1 , j 0 , and 0 , where g 3 []a + q ()A + B 2 w()L,,,eqh = 3 + 1 - - j j j j () 1 å * . d0 ()e+h []1 2 41()q- e k d 1 L j = 1 j j 0 q = × ()+ -1 We choose j aj Aj Bj and obtain that g 1 K1 w ()L,,,eqh = 3 + - 1 - . d ()1 2 ()1 - e d 0 ()e1 + h []L 0 It is easy to see that if (6.10) is valid, then there exist 0 < 0 such that w ()L,,,eqh > 0 . Therefore, equation (6.21) gives us that

3 ()2 + q y ()2 £ wt å j j t j = 1 t æö3 £ ç÷2 + q y 2 × -w*t + × -w*()t -t ()()t 2 t w0 å j j 0 e CL, h e max lj w d , èøò j j = 1 0 where w* > 0 . Thus, if (6.10) is valid, then equation (6.14) implies (6.15). Determining Functionals in the Problem of Nerve Impulse Transmission 345

Let us now assume that (6.11) is valid. Since -g()()¶, * , 2 £ -g¶ 2 + æö()+ g × + (), * × ¶2 GU U x w 3 xw èø1 2 w hU U x w , then it follows from (6.16) that d 1 × d ¶ w 2 ++0 ¶2 w 2 g ¶ w 2 £ 2 dt x 2 x 3 x 3 a2 £ 2 ××()g + g 2 2 + × j × y 2 1 2 w 2 å j . (6.22) d0 d j = 1 0 Therefore, we can use equation (6.18) and obtain (cf. (6.19)) that

~~~~æö3 d 3 1 × d ç÷¶ w 2 + q y 2 +++0 ¶2 w 2 g ¶ w 2 e k*q y 2 £ 2 dt ç÷x å j j 2 x 3 x å j j j èøj = 1 j +1~~~~

æö3 a2 ()A + B 2 £ ç÷2 × ()g + g 2 + 9 × j j j × 2 ç÷1 2 å w d0 4 ()- e 2 []* 2 èøj = 1 1 kj d0 < 0 . Therefore, æö3 3 1 ×gd ç÷¶ 2 + q y 2 ++¶ 2 e * q y 2 £ × ()2 x w å j j 3 x w å kj j j CL, h max lj w , 2 dt ç÷ j èøj = 1 j + 1 provided that 3 9 a2 ()A + B 2 1 - - 4 × ()g + g 2 - j j j ³ 1 1 2 å 0 . ()2 2 d2 ()- e 2 []* 2 2 ()e1+h []L 0 j = 1 21 kj d0 As in the first part of the proof, we can now conclude that if (6.11) is valid, then (6.14) implies the equations ¶ () = y () = = ,, limx wt 0 andlimj t 0 ,j 123 . (6.23) t ® ¥ t ® ¥ It follows from (6.17) that 3 d 2 + g 2 £ 3 2 y ()2 w 3 w g å aj j t . dt 3 j = 1 Therefore, as above, we obtain equation (6.15) for the case (6.11). Theorem 6.1 is proved. 346 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C h As in the previous sections, modes, nodes and generalized local volume averages can a be choosen as determining functionals in Theorem 6.1. p t e {} 2(), Exercise 6.1 Let ek be a basis in L 0 L which consists of eigenvec- r ¶2 tors of the operator x with one of the boundary conditions (6.5). 5 Show that the set L ìü L = íýl ()u = e ()x ux()dx , j = 1,,¼ N îþj ò j 0 is determining (in the sense of (6.14) and (6.15)) for problem (6.1)– (6.5) for N large enough.

Exercise 6.2 Show that in the case of the Neumann boundary conditions (6.5b) it is sufficient to choose the number N in Exercise 6.1 such that K L j N > p ,.(6.24)j = 1 or 2 d0 Obtain a similar estimate for the other boundary conditions (Hint: see Exercises 3.2–3.4).

Exercise 6.3 Let ìü L = íýl : l ()u = ux(), x = jh, h = L , j = 1,,¼ N .(6.25) îþj j j j N Î Show that for every wV1 the estimate 2 2 2 2 ¶ w ³ w - C , h max l ()w (6.26) x ()1 +h h2 N j holds for any h > 0 (Hint: see Exercise 3.6).

Exercise 6.4 Use estimate (6.26) instead of (6.20) in the proof of Theorem 6.1 to show that the set of functionals (6.25) is determining for prob- > × ()¤ lem (6.1)–(6.5), provided that NL 2 K1 d0 . Exercise 6.5 Obtain the assertions similar to those given in Exercises 6.3 and 6.4 for the following set of functionals h ì t L = ()= 1 l æö ()t+ t í lj w èøuxj d , î h ò h 0 ü x = jh, h = L , j = 0,,¼ N -1ý , j N þ Determining Functionals in the Problem of Nerve Impulse Transmission 347

~~~~where the function l ()x Î L¥ ()R1 possesses the properties ¥ ò l()x dx = 1 ,.supp l()x Ì []01, -¥~~~~

It should be noted that in their work Hodgkin and Huxley used the following expres- () () sions (see [13]) for kj u and hj u : a () j u k ()u = a()u + b()u ,,h ()u = j j j j a ()+ b() j u j u a ()=b()- + ()= ()- ¤ where 1 u e 0.1 u 2.5 ,1 u 4 exp u 18 ; 0.07 1 a ()u =b ,;()u = 2 e ()-0.05 u 2 10.1+ exp()- u + 3 a ()=b()- + ()= ()- ¤ 3 u 0.1 e 0.1 u 1 ,.3 u 0.125 exp u 80 ()=d¤ ()z - =d=g- = Here ez ze 1 . They also supposed that 1 115 , 2 12 , 1 120 , g = £ × 4 £ × 4 and 2 36 . As calculations show, in this case K1 5.2 10 and K2 7.4 10 . { = = ¤ =},,,¼ , Therefore (see Exercise 6.4), the nodes xj jh, hlN, j 012 N ³ ××2 ¤ are determining for problem (6.1)–(6.5) when N 2.3 10 Ld0 . Of course, similar estimates are valid for modes and generalized volume averages. Thus, for the asymptotic dynamics of the system to be determined by a small ¤ number of functionals, we should require the smallness of the parameter Ld0 . However, using the results available (see [14]) on the analyticity of solutions to problem (6.1)–(6.5) one can show (see Theorem 6.2 below) that the values of all = (),,, components of the state vector Uuv1 v2 v3 in two sufficiently close nodes uniquely determine the asymptotic dynamics of the system considered not depend- ¤ ing on the value of the parameter Ld0 . Therewith some regularity conditions for the coefficients of equations (6.1) and (6.2) are necessary. Let us consider the periodic initial-boundary value problem (6.1)–(6.5c). As- > () () sume that dj 0 for all j and the functions kj u and hj u are polynomials such ()> ££() Î []d , d that kj u 0 and 0 hj u 1 for u 2 1 . In this case (see [14]) every solution ()= ()(), ,,,(), (), (), Ut ux t v1 xt v2 xt v3 xt > possesses the following Gevrey regularity property: there exists t* 0 such that ¥ æö3 ç÷()() 2 + ()() 2 × t l £ å ç÷Fl ut å Fl vj t e C (6.27) l = -¥ èøj = 1 t > ³ () for some 0 and for all tt* . Here Fl w are the Fourier coefficients of the function wx() : 348 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C L h ìü a 1 2 p l F ()w = × wx()× exp íýi x dx , l = 01,,,± ±¼2 p l L ò L t îþ e 0 r In particular, property (6.27) implies that every solution to problem (6.1)–(6.5c) be- 5 comes a real analytic function for all t large enough. This property enables us to prove the following assertion.

Theorem 6.2. > () () Let dj 0 for all jk and let j u and hj u be the polynomials possessing the properties ()> ££() Î []d , d kj u 0 ,,for0 hj u 1 for u 2 1 . £ < £ Let x1 and x2 be two nodes such that 0 x1 x2 L and - < ¤ x2 x1 2 d0 K1 , where K1 is defined by formula (6.12). Then for every two solutions ()= ()(), ,,,(), (), (), Ut ux t v1 xt v2 xt v3 xt and *()= ()*(), ,,,* (), * (), * (), U t u xt v1 xt v2 xt v3 xt to problem (6.1)–(6.5 c) the condition ìü3 (), - *(), + (), - *(), = limmax íýuxl t u xl t å vj xl t vj xl t 0 t ® ¥ l = 12, îþ j = 1

~~~~implies their asymptotic closeness in the space H0 : ìü3 ()- *()2 + ()- *()2 = limíýut u t å vj t vj t 0 . (6.28) t ® ¥ îþ j = 1~~~~

~~~~Proof. =y- * = - * = ,, Let wuu and let j vj vj , j 123 . We introduce the notations: x2 D =D{}££ = - 2 = ()2 x : x1 xx2 ,,and.x2 x1 w D ò wx dx~~~~

~~~~x1 Let ìü3 (), D = (), + y(), mt max íýwxl t å j xl t . l = 12, îþ j = 1 As in the proof of Theorem 6.1, it is easy to find that~~~~

~~~~1 × d w 2 ++£d ¶ w 2 g w 2 2 dt D 0 x D 3 D Determining Functionals in the Problem of Nerve Impulse Transmission 349~~~~

~~~~3 £ ()¶, × (), + y × å wxl t x wxl t å aj j D w D l = 12, j = 1 and~~~~

1 × d y 2 +yk* y 2 £ ()¶x , t × y ()x , t + ()yA + B × w . 2 dt j D j j D å j l x j l j j j D D l = 12, It follows from (6.27) that ìü3 ¶ (), + ¶y (), < ¥ sup max íýx wx t å x j xt . t > 0 x Î []0, L îþ j = 1 <

~~~~£ ××()- e -1 2 + (), D K1 1 w D mt , where K1 is defined by formula (6.12). Simple calculations give us that~~~~

x2 x2 D 2 w 2 º wx()2 dx £ ()1+h × ¶ wx()2 dx + C ××D wx()2 D ò 2 ò x h 1 x1 x1 h >D2 < ¤ < 0 such that æö3 æö3 d ç÷w 2 + q y 2 + w × ç÷w 2 + q y 2 £ Cmt× (), D , dt ç÷D å j j D ç÷D å j j D èøj = 1 èøj = 1 where w is a positive constant. As in the proof of Theorem 6.1, it follows that condition mt(), D ® 0 as t ® ¥ implies that æö3 ()- * º ç÷2 + q y 2 = lim Ut U t D lim w D å j j D 0 .(6.29) t ® ¥ t ® ¥ èø j = 1 Let us now prove (6.28). We do it by reductio ad absurdum. Assume that there exists ® ¥ a sequence tn + such that ()- *> limUtn U tn 0 .(6.30) n ® ¥ {} {}* A Let Vn and Vn be sequences lying in the attractor of the dynamical system generated by equations (6.1)–(6.5c) and such that ()- ® *- * ® Utn Vn 0 ,.(6.31)U tn Vn 0 350 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C {} h Using the compactness of the attractor we obtain that there exist a sequence nk * * * a and elements VV, ÎA such that Vn ® V and Vn ® V . Since p k k t - * £ ()- * ++()- *- * e Vn Vn D Utn U tn D Utn Vn U tn Vn , r it follows from (6.29) and (6.31) that 5 - * ==- * VVD limVn Vn 0 . k ® ¥ k k D Therefore, Vx()= V*x for x Î D . However, the Gevrey regularity property implies that elements of the attractor are real analytic functions. The theorem on the uniqueness of such functions gives us that Vx()º V*x for x Î []0, L . Hence, V - V * ® 0 as k ® ¥ . Therefore, equation (6.31) implies that nk nk ()- *()= limUtn U tn 0 . k ® ¥ k k This contradicts assumption (6.30). Theorem 6.2 is proved.

It should be noted that the connection between the Gevrey regularity and the existence of two determining nodes was established in the paper [15] for the first time. The results similar to Theorem 6.2 can also be obtained for other equations (see the references in [3]). However, the requirements of the spatial unidimensionality of the problem and the Gevrey regularity of its solutions are crucial.

~~~~§ 7 Determining Functionals for Second Order in Time Equations~~~~

In a separable Hilbert space H we consider the problem \\ \ \ u ++g u Au = Bu(), t ,,,(7.1)u = u u = u ts= 0 ts= 1 where the dot over utA stands for the derivative with respect to , is a positive operator with discrete spectrum, g > 0 is a constant, and Bu(), t is a continuous mapping from DA()q ´ R into H with the property (), - (), £ ()r × q()- Bu1 t Bu2 t M A u1 u2 (7.2) £ q < ¤ Î ()12 12 £ r for some 0 12 and for all uj DA such that A uj . Assume Î Î ()12 Î that for any s R , u0 DA , and u1 H problem (7.1) is uniquely solvable in the class of functions Cs()[ , + ¥); DA()12 Ç C1 ()[s,,+ ¥) H (7.3) and defines a process ()H ; St(), t in the space H = DA()12 ´ H with the evolutionary operator given by the formula (), (); = ()(); \ () St s u0 u1 ut u t ,(7.4) Determining Functionals for Second Order in Time Equations 351 where ut() is a solution to problem (7.1) in the class (7.3). Assume that the process ()H ; St(), t is pointwise dissipative, i.e. there exists R > 0 such that (), £ ³ + () St sy0 H R ,(7.5)tst0 y0 = (); Î H for all initial data y0 u0 u1 . The nonlinear wave equation (see the book by A. V. Babin and M. I. Vishik [8]) ì ï ¶2u ¶u + g - Du = fu(), x Î W , t > 0 , ï ¶ 2 ¶t í t ï ¶u u ==0, u u ()x , =u ()x , ï ¶W t = 0 0 ¶t 1 î t = 0 is an example of problem (7.1) which possesses all the properties listed above. Here W is a bounded domain in Rd and the function fu()Î C1()R possesses the properties: u 1 -()l - e u2 - C ££fv()dv C uf() u ++C ()l - e u2 , 1 1 ò 2 3 2 1 0 ¢()£ ()+ b f u C4 1 u , l -D where 1 is the first eigenvalue of the operator with the Dirichlet boundary ¶W >e>b£ ()- -1 ³ conditions on , Cj 0 and 0 are constants, 2 d 2 for d 3 and b is arbitrary for d = 2 .

Theorem 7.1. L = {}= ,,¼ Let lj : j 1 N be a set of continuous linear functionals onDA()12 . Assume that g 1 12 12- q eeº L ()DA(); H < , (7.6) ()()+ l-1 g2 32 MR 441 where RM is the radius of dissipativity (see (7.5)), ()r and qÎ [012, ¤) l are the constants from (7.2), and 1 is the first eigenvalue of the operator A. Then L is an asymptotically determining set for problem (7.1) in the () () sense that for a pair of solutions u1 t and u2 t from the class (7.3) the condition t +1 ()t()t - ()t = = ,,¼ limlj u1 u2 d 0 for j 1 N (7.7) t ® ¥ ò t implies that ìü 12 ()()- () 2 + \ ()- \ ()2 = limíýA u1 t u2 t u1 t u2 t 0 . (7.8) t ® ¥ îþ 352 Theory of Functionals that Uniquely Determine Long-Time Dynamics

~~~~C h Proof. a We rewrite problem (7.1) in the form p t dy e +A y = B ()yt, ,,y = y (7.9) r dt ts= 0 5 where \ \ \ yuu= (); ,,A yu= ()- ; gu + Au B ()yt, = ()0 ; Bu(), t .~~~~

Lemma 7.1. There exists an exponential operator exp {}-tA in the space H = = DA()12 ´ H and g2 ìül g t exp {}-tA y £ 21+ × expíý-1 y , (7.10) H l l + g2 H 1 îþ4 1 2 l where 1 is the first eigenvalue of the operator A .

Proof. = () ()==-tA ()(); \ () Let y0 u0 ; u1 . Then it is evident that yt e y0 ut u t , where ut() is a solution to the problem \\ \ \ u ++gu Au = 0 ,,,(7.11)u = u u = u t = 0 0 t = 0 1 (see Section 3.7 for the solvability of this problem and the properties of solutions). Let us consider the functional æö\ g Vu()= 1 æöu 2 + A12 u 2 + n uu, + u ,,0 <

~~~~d Vut()() = -n() gn- u\ 2 - A12 u 2 .(7.13) dt Therefore, it follows from (7.12) and (7.13) that~~~~

d Vut()b() + Vut()() £ dt æöæön æöæö- £ -ngn- - 1 + b u\ 2 - - 1 + l 1ng b A12 u 2 . èøèø2 2g èøèø2 1 n =b()g¤ = ()+ l-1 g 2 -1 × g Hence, for 12 and 2 1 we obtain that Determining Functionals for Second Order in Time Equations 353

~~~~d Vut()b() + Vut()() £ 0 , dt~~~~

~~~~æöæö- æö 1 u\ 2 + A12 u 2 ££Vu() 3 + l 1 g 2 u\ 2 + A12 u 2 . 4 èøèø4 1 èø This implies estimate (7.10). Lemma 7.1 is proved.~~~~

It follows from (7.9) that t -()- A -()-t A yt()= e ts ys()+ òe t B()ty()tt , d . s ()= Therefore, with the help of Lemma 7.1 for the difference of two solutions yj t = ()(); \ () = , uj t uj t , j 12 , we obtain the estimate t () £ -b ()ts- () + -b ()t -t ()()t -t()()t yt H De ys H Deò Bu1 Bu2 d ,(7.14) s ()= ()- () b where yt y1 t y2 t and the constants D and have the form

- gl D = 21+ l 1 g2 ,.b = 1 1 l + g 2 4 1 2 () £ ³³ By virtue of the dissipativity (7.5) we can assume that yj t H R for all tss0 . Therefore, equations (7.14) and (7.2) imply that t () £ -b ()ts- () + () -b ()t - t q()t()t - ()t yt H De ys H DM R ò e A u1 u2 d . s The interpolation inequality (see Exercise 2.1.12) - q q Aq uu£ 12 A12 u 2 ,,0 ££q 1 2 Theorem 2.1, and the result of Exercise 2.12 give us that q £ ()+ e12- q 12 A uCL max lj u L A u , j 12 where eL = eL ()DA(), H . Therefore, t () £ -b ()ts- () ++()e× 12- q -b ()t - t ()t t yt H De ys H DM R L ò e y H d s t + ()L,, × -b ()t - t ()t()t C DR ò e NL u d , s = {}() = ,,¼ , where NL u max lj u : j 12 N . If we introduce a new unknown b t function y()t = e yt()H in this inequality, then we obtain the equation 354 Theory of Functionals that Uniquely Determine Long-Time Dynamics

~~~~C h t t a y()t £ Dy()s ++ a yt()td CebtN ()tu()t d , p ò ò L t s s e - q r 12 where a = DM()e R L . We apply Gronwall’s lemma to obtain~~~~

5 t ìüt ïï a ()ts- a t -at bx y()t £ Dy()s e + Ce e íýe NL ()xu()x d dt . ò ïïò s îþs After integration by parts we get t () £ () {}-()ba- ()- + -()ba- ()t - t ()t()t yt H Dys H exp ts Ceò NL u d . s If equation (7.6) holds, then wba= - > 0 . Therewith it is evident that for 0 < 0 . This implies (7.8). Theorem 7.1 is proved.

Unfortunately, because of the fact that condition (7.2) is assumed to hold only for 0 £ q < 12¤ , Theorem 7.1 cannot be applied to the problem on nonlinear plate oscillations considered in Chapter 4. However, the arguments in the proof of Theo- rem 7.1 can be slightly modified and the theorem can still be proved for this case using the properties of solutions to linear nonautonomous problems (see Section 4.2). However, instead of a modification we suggest another approach (see also [3]) which helps us to prove the assertions on the existence of sets of determining functionals for second order in time equations. As an example, let us consider a problem of plate oscillations . Determining Functionals for Second Order in Time Equations 355

~~~~Thus, in a separable Hilbert space H we consider the equation~~~~

~~~~ì \\ \ æö ï u ++gu A2uMA +12 u 2 Au + Lu = pt() , (7.15) í èø \ ï u ==u , u u . (7.16) î t = 0 0 t = 0 1~~~~

We assume that A is an operator with discrete spectrum and the function Mz() lies 1() in C R+ and possesses the properties: z a) M ()z º òM()x d x³ - az- b , (7.17) 0 £ < l Îl where 0 a 1 , b R , and 1 is the first eigenvalue of the operator A ; > b) there exist numbers aj 0 such that ()- M ()³ 1 + a - ³ zM z a1 z a2 z a3 ,(7.18)z 0 with some constant a > 0 . We also require the existence of 0 £ q < 1 and C > 0 such that Lu£ C Aq u ,.uDAÎ ()q (7.19) These assumptions enable us to state (see Sections 4.3 and 4.5) that if Î () Î ()Îg¥ (), > u0 DA,,u1 H pt L R+ H ,,(7.20)0 then problem (7.15) and (7.16) is uniquely solvable in the class of functions W = (); ()Ç 1(); C R+ DA C R+ H .(7.21) Therewith there exists R > 0 such that ()2 + \ ()2 £ 2 ³ (), Au t u t R ,(7.22)tt0 u0 u1 for any solution ut()ÎW to problem (7.15) and (7.16).

Theorem 7.2. L = {}= ,,¼ , Assume that conditions (7.17)–(7.20) hold. Let lj : j 12 N () e> be a set of continuous linear functionals on DA. Then there exists 0 0 depending both on R and the parameter of equation (7.15) such that the eeº ()e(); < L condition L DA H 0 implies that is an asymptotically determining set of functionals with respect to DA()´ H for problem (7.15) and W () () (7.16) in the class of solutions , i.e. for two solutions u1 t and u2 t from W the condition t +1 ()()t - ()t 2 t = Î L limlj u1 u2 d 0 , lj (7.23) t ® ¥ ò t implies that 356 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C ìü h \ ()- \ ()2 + ()()- () 2 = a limíýu1 t u2 t Au1 t u2 t 0 . (7.24) p t ® ¥ îþ t e r Proof. () () W 5 Let u1 t and u2 t be solutions to problem (7.15) and (7.16) lying in . Due to equation (7.22) we can assume that these solutions possess the property ()2 + \ ()2 £ 2 ³ = , Auj t uj t R ,,t 0 j 12 .(7.25) ()= ()- () Let us consider the function ut u1 t u2 t as a solution to equation \\ ++g \ 2 +æö12 ()2 = ()(), () u u A uMAèøu1 t Au Fu1 t u2 t ,(7.26) where

~~~~(), = æö12 2 - æö12 2 - ()- Fu1 u2 MAèøu2 MAèøu1 Au2 Lu1 u2 . It follows from (7.19) and (7.25) that~~~~

~~()(), () £ æö12 + q Fu1 t u2 t CR èøA uAu .(7.27) Let us consider the functional \ \ ìü\ g Vu(), u; t = 1 Euu()n, ; t + íý()uu, + u 2 (7.28) 2 îþ2 on the space H = DA()´ H , where~~

(), \ ; = \ 2 ++2 æö12 ()2 12 2 +m 2 Euu t u Au MAèøu1 t A u u and the positive parameters mn and will be chosen below. It is clear that for ()uu; \ Î H we have (), \ ; ³ \ 2 ++2 12 2 +m 2 Euu t u Au mR A u u , = {}() ££l-1 2 where mR min Mz: 0 z 1 R . Moreover, g - 1 u\ 2 ££()uu, \ + u\ 2 1 u\ 2 + g u 2 . 2g 2 2g Therefore, the value m can be chosen such that

~~~~a æö2 + \ 2 ££(), aæö2 + \ 2 1 èøAu u Vu t 2 èøAu u (7.29) for all 0 <~~~~

~~~~With the help of (7.25) and (7.27) we obtain that \ g \ æö 1 d Eu(), u; t £ - u()t 2 + C A12 u 2 + Aqu 2 . 2 dt 2 R èø Using (7.26) and (7.27) it is also easy to find that~~~~

~~~~ìü\ g \\ \ d íý()uu, + ()uu, = u\ 2 + ()uu,g+ u £ dt îþ2~~~~

~~~~£ \ 2 - 2 ++12 2 æö12 2 + q 2 u Au MR A u CR èøA u A u u , where~~~~

ìü- M = max íýMz(): 0 ££z l 1R2 . R îþ1 We choose ng= ¤ 4 and use the estimate of the form b b 1- b A uAu££×eu Au+ Ce u ,,,0 < 0 to obtain that ìüg d (), \ ; = d 1 ()n, \ ; + æö, \ + £ Vu u t íýEu u t èøuu u dt dt îþ2 2 g æö £ - Au 2 + u\ 2 + C ut()2 . 8 èøR Therefore, using the estimate ()2 £ ()2 + e ()(), 2 ut CL maxlj u 2 L DA H Au j and equation (7.29) we obtain the inequality d ()w(), \ (); + ()(), \ (); £ ()() 2 Vut u t t Vut u t t Clmax j ut , dt j g - provided e ()eDA(), H < =w C 1 . Here is a positive constant. As above, this L 0 16 R easily implies (7.24), provided (7.23) holds. Theorem 7.2 is proved.

~~~~Exercise 7.1 Show that the method used in the proof of Theorem 7.2 also enables us to obtain the assertion of Theorem 7.1 for problem (7.1).~~~~

~~Exercise 7.2 Using the results of Section 4.2 related to the linear variant of equation (7.15), prove that the method of the proof of Theorem 7.1 can also be applied in the proof of Theorem 7.2.~~

Thus, the methods presented in the proofs of Theorems 7.1 and 7.2 are close to each other. The same methods with slight modifications can also be used in the study of problems like (7.1) with additional retarded terms (see [3]). 358 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C h Exercise 7.3 Using the estimates for the difference of two solutions to equ- a p ation (7.15) proved in Lemmata 4.6.1 and 4.6.2, find an analogue t of Theorems 1.3 and 4.4 for the problem (7.15) and (7.16). e r 5

~~~~§ 8 On Boundary Determining Functionals~~~~

The fact (see Sections 5–7 as well as paper [3]) that in some cases determining functionals can be defined on some auxiliary space admits in our opinion an interesting generalization which leads to the concept of boundary determining functionals. We now clarify this by giving the following simple example. In a smooth bounded domain W Ì Rd we consider a parabolic equation with the nonlinear boundary condition

ì ¶u ï = n Du - fu(), x Î W , t > 0 , ï ¶t í (8.1) ï ¶u + hu() ==0, u u ()x . ï ¶ ¶W t = 0 0 î n ¶W Assume that n is a positive parameter, fz() and hz() are continuously differentiable functions on R1 such that f ¢()z ³ - a,,h¢()z £ b (8.2) where a ³b0 and > 0 are constants. Let W = 21, ()W ´ Ç 10, ()W ´ C R+ C R+ .(8.3) 21, ()W ´ (), W´ Here C R+ is a set of functions uxt on R+ that are twice continuously differentiable with respect to x and continuously differentiable with respect 10, ()W ´ to tC . The notation R+ has a similar meaning, the bar denotes the closure of a set. (), (), Let u1 xt and u2 xt be two solutions to problem (8.1) lying in the class W (we do not discuss the existence of such solutions here and refer the reader to ()= ()- () the book [7]). We consider the difference ut u1 t u2 t . Then (8.1) evidently implies that

~~~~1 d ut()2 ++nÑut()2 ()fu()()t - fu()()t , ut() = 2 dt L2()W L2()W 1 2 L2 ()W~~~~

~~~~= -n ()()s()() - ()() ò ut hu1 t hu2 t d . ¶W On Boundary Determining Functionals 359~~~~

~~~~Using (8.2) we obtain that~~~~

1 d ut()2 +nbnÑut()2 - a ut()2 £ × ut()2 .(8.4) 2 dt L2 ()W L2()W L2()W L2()¶W W One can show that there exist constants c1 and c2 depending on the domain only and such that æö u 2 £ c u 2 + Ñu 2 ,(8.5) L2()W 1èøL2()¶W L2()W æö u 2 £ c u 2 + Ñu 2 .(8.6) H12 ()¶W 2èøL2()W L2()W Here Hs ()¶W is the Sobolev space of the order s on the boundary of the domain W . Equations (8.4)–(8.6) enable us to obtain the following assertion.

Theorem 8.1. L = {}= ,,¼ Let lj : j 1 N be a set of continuous linear functionals 12 ()¶W a< n on the space H . Assume that c1 and na- c 12 e º e ()H12 ()¶W , L2()¶W < 1 º e , (8.7) L L n ()+ ()+ b 0 1 c1 c2 1 nab,,, where the constants c1 , and c2 are defined in equations (8.1), (8.2), (8.5), and (8.6). Then L is an asymptotically determining set with respect to L2()W for problem (8.1) in the class of classical solutions W .

Proof. ()= ()- () ()Î W Let ut u1 t u2 t , where uj t are solutions to problem (8.1). Theorem 2.1 implies that 2 £ ()2 + ()e+ d 2 2 u 2 CL , d max lj u 1 L u 12 (8.8) L ()¶W j H ()W for any d > 0 . Equations (8.5) and (8.6) imply that 2 £ ()2 + ()+ ()e+ d 2 æö2 + Ñ 2 u 2 CL , d maxl u 1 c c 1 L u 2 u 2 . L ()¶W j j 1 2 èøL ()¶W L ()W Therefore, equation (8.4) gives us that

1 d u 2 + n u 2 +aÑu 2 - ut()2 £ 2 dt L2()W L2()¶W L2()W L2()W æö2 2 £ n()1+b ut() £ n ()1 + c c ()1 + d ()e1 +b 2 u + Ñu + L2()¶W 1 2 L èøL2()¶W L2()W + ()2 Clmax j u . j Using estimate (8.5) once again we get ìüc 1 d 2 + n -a()+ ()+ d ()e+b 2 - 1 2 £ ()2 u íý11c1 c2 1 1 L n u Clmax j u , (8.9) 2 dt c1 îþj 360 Theory of Functionals that Uniquely Determine Long-Time Dynamics

C h provided that a c p 11-a()+ c c ()1 + d ()e1 + b 2 - 1 > 0 .(8.10) t 1 2 L n e r It is evident that (8.10) with some d > 0 follows from (8.7). Therefore, inequality 5 (8.9) enables us to complete the proof of the theorem. Thus, the analogue of Theorem 3.1 for smooth surfaces enables us to state that problem (6.1) has finite determining sets of boundary local surface averages.

An assertion similar to Theorem 8.1 can also be obtained (see [3]) for a nonlinear wave equation of the form ¶2u ¶u + g = Du - fu(),,,x Î W Ì Rd t > 0 ¶t2 ¶t ¶u ¶u ¶u = -ja - æöu ,,u = 0 u = u ()x , = u ()x . ¶n ¶t èøG ¶W\ G t = 0 0 ¶t 1 G G t = 0 Here GW is a smooth open subset on the boundary of , fu() and j()u are bounded continuously differentiable functions, and ag and are positive parameters. References

1. FOIAS C., PRODI G. Sur le comportement global des solutions non stationnaires des equations de Navier-Stokes en dimension deux // Rend. Sem. Mat. Univ. Padova. –1967. –Vol. 39. –P. 1–34. 2. LADYZHENSKAYA O. A. A dynamical system generated by Navier-Stokes equations // J. of Soviet Mathematics. –1975. –Vol. 3. –P. 458–479. 3. CHUESHOV I. D. Theory of functionals that uniquely determine the asymptotic dynamics of infinite-dimentional dissipative systems // Russian Math. Surveys. –1998. –Vol. 53. –#4. –P. 731–776. 4. LIONS J.-L., MAGENES E. Non-homogeneous boundary value problems and applications. –New York: Springer, 1972. –Vol.1. 5. MAZ’YA V. G. Sobolev spaces. –Leningrad: Leningrad Univ. Press, 1985 (in Russian). 6. TRIEBEL H. Interpolation theory, function spaces, differential operators. –Amsterdam: North-Holland, 1978. 7. LADYZHENSKAYA O. A., SOLONNIKOV V. A., URALTCEVA N. N. Linear and quasilinear equations of parabolic type. –Providence: Amer. Math. Soc., 1968. 8. BABIN A. V., VISHIK M. I. Attractors of evolution equations.–Amsterdam: North-Holland, 1992. 9. TEMAM R. Infinite-dimensional dynamical systems in Mechanics and Physics. –New York: Springer, 1988. 10. SMOLLER J. Shock waves and reaction-diffusion equations.–New York: Springer, 1983. 11. TEMAM R. Navier-Stokes equations, theory and numerical analysis. 3rd edn. –Amsterdam: North-Holland, 1984. 12. FOIAS C., MANLEY O. P., TEMAM R., TREVE Y. M. Asymptotic analysis of the Navier-Stokes equations // Physica D. –1983. –Vol. 9. –P. 157–188. 13. HASSARD B. D., KAZARINOFF N. D., WAN Y.-H. Theory and application of Hopf bifurcation. –Cambridge: Cambridge Univ. Press, 1981. 14. PROMISLOW K. Time analyticity and Gevrey regularity for solutions of a class of dissipative partial differential equations // Nonlinear Anal., TMA. –1991. –Vol. 16. –P. 959–980. 15. KUKAVICA I. On the number of determining nodes for the Ginzburg-Landau equation // Nonlinearity. –1992. –Vol. 5. –P. 997–1006.

~~~~Chapter 6~~~~

~~~~Homoclinic Chaos in Infinite-Dimensional Systems~~~~

~~~~Contents~~~~

. . . . § 1 Bernoulli Shift as a Model of Chaos ...... 365 . . . . § 2 Exponential Dichotomy and Difference Equations ...... 369 . . . . § 3 Hyperbolicity of Invariant Sets for Differentiable Mappings ...... 377 . . . . § 4 Anosov’s Lemma on e -trajectories ...... 381 . . . . § 5 Birkhoff-Smale Theorem ...... 390 . . . . § 6 Possibility of Chaos in the Problem of Nonlinear Oscillations of a Plate ...... 396 . . . . § 7 On the Existence of Transversal Homoclinic Trajectories . . 402

~~~~. . . . References ...... 413~~~~

In this chapter we consider some questions on the asymptotic behaviour of a discrete dynamical system. We remind (see Chapter 1) that a discrete dynamical system is defined as a pair ()XS, consisting of a metric space X and a continuous mapping of X into itself. Most assertions on the existence and properties of attractors given in Chapter 1 remain true for these systems. It should be noted that the following examples of discrete dynamical systems are the most interesting from the point of view of applications: a) systems generated by monodromy operators (period mappings) of evolutionary equations, with coefficients being periodic in time; t -1()=- b) systems generated by difference schemes of the type un +1 un = () = ,,,¼ Fun , n 012 in a Banach space X (see Examples 1.5 and 1.6 of Chap- ter 1). The main goal of this chapter is to give a strict mathematical description of one of the mechanisms of a complicated (irregular, chaotic) behaviour of trajectories. We deal with the phenomenon of the so-called homoclinic chaos. This phenomenon is well-known and is described by the famous Smale theorem (see, e.g., [1–3]) for finite-dimensional systems. This theorem is of general nature and can be proved for infinite-dimensional systems. Its proof given in Section 5 is based on an infinite-dimensional variant of Anosov’s lemma on e -trajectories (see Section 4). The considerations of this Chapter are based on the paper [4] devoted to the finite-dimensional case as well as on the results concerning exponential dichotomies of infinite-dimensional systems given in Chapter 7 of the book [5]. We follow the arguments given in [6] while proving Anosov’s lemma.

~~~~§ 1 Bernoulli Shift as a Model of Chaos~~~~

Mathematical simulation of complicated dynamical processes which take place in real systems requires that the notion of a state of chaos be formalized. One of the possible approaches to the introduction of this notion relies on a selection of a class of explicitly solvable models with complicated (in some sense) behaviour of trajectories. Then we can associate every model of the class with a definite type of chaotic behaviour and use these models as standard ones comparing their dynamical structure with a qualitative behaviour of the dynamical system considered. A discrete dynamical system known as the Bernoulli shift is one of these explicitly solvable models. Let m ³ 2 and let ìü S = íýx = ()¼,x- ,,,x x ¼ : x Î {}12,,¼ ,m , j ÎZ , m îþ1 0 1 j S i.e. m is a set of two-sided infinite sequences the elements of which are the inte- ,,¼ , S gers 12 m . Let us equip the set m with a metric 366 Homoclinic Chaos in Infinite-Dimensional Systems

C ¥ - h xi yi a (), = - i dxy å 2 + - .(1.1) p 1 xi yi t i = -¥ e = {}Î =S{}Î r Here xxi : i Z and yyi : i Z are elements of m . Other methods of introduction of a metric in S are given in Example 1.1.7 and Exercise 1.1.5. 6 m Exercise 1.1 Show that the function dxy(), satisfies all the axioms of ametric. = {} =S{} Exercise 1.2 Let xxi and yyi be elements of the set m . = £ Assume that xi yi for iN and for some integer N . Prove that dxy(), £ 2-N + 1 . (), < -N ,SÎ Exercise 1.3 Assume that equation dxy 2 holds for xy m , = £ - where Nx is a natural number. Show that i yi for all iN1 (), ³ - i -1 ¹ (Hint: dxy 2 ifxi yi ). Î S Exercise 1.4 Let x m and let U N()= {}Î S = £ x y m : xi yi for iN. (1.2) Prove that for any 0 <

S In the space m we define a mapping S which shifts every sequence one symbol left, i.e. []= Î = {}Î S Sx i xi +1 ,,i Z xxi m . Evidently, S is invertible and the relations dSx(), Sy £ 2 dxy(), , dS()-1xS, -1y £ 2 dxy(), ,SÎ hold for all xy m . Therefore, the mapping S is a homeomorphism. ()S , The discrete dynamical system m S is called the Bernoulli shift of the space of sequences of m symbols. Let us study the dynamical properties of the sys- ()S , tem m S . ()S , Exercise 1.6 Prove that m S has m fixed points exactly. What structure do they have? Bernoulli Shift as a Model of Chaos 367

=a()a ,,¼a Î {},,¼ We call an arbitrary ordered collection a 1 N with j 1 m Î S a segment (of the length Nx ). Each element m can be considered as an or- S dered infinite family of finite segments while the elements of the set m can be con- = ()a ,,¼a structed from segments. In particular, using the segment a 1 N we can Î S construct a periodic element a m by the formula = a Î {},,¼ Î aNk+ j j ,,.(1.3)j 1 N k Z

= ()a ,,¼a Exercise 1.7 Let a 1 N be a segment of the length N and let Î S a m be an element defined by (1.3). Prove that a is a periodic ()S , point of the period N of the dynamical system m S , i.e. SN aa= .

Exercise 1.8 Prove that for any natural N there exists a periodic point of the minimal period equal to N . S Exercise 1.9 Prove that the set of all periodic points is dense in m , i.e. for ÎeS > every x m and 0 there exists a periodic point a with the property dxa()e, < (Hint: use the result of Exercise 1.4).

~~~~Exercise 1.10 Prove that the set of nonperiodic points is not countable.~~~~

Exercise 1.11 Let a = ()¼aaa¼,,,, and b = ()¼bbb¼,,,, be fi- ()S , = {} xed points of the system m S . Let Cci be an element S = a £ - = b ³ ofm such that ci for iM1 and ci for iM2 , where M1 and M2 are natural numbers. Prove that lim Sn c = a ,.(1.4)lim Sn c = b n ® -¥ n ® ¥

Î S ¹ ¹ Assume that an element c m possesses property (1.4) with ca andcb . Ifab¹ , then the set g = {}n Î ab, S c : n Z is called a heteroclinic trajectory that connects the fixed points ab and . =g= g Ifab , then a aa, is called a homoclinic trajectory of the point a . The elements of a heteroclinic (homoclinic, respectively) trajectory are called heteroclinic (homoclinic, respectively) points.

Exercise 1.12 Prove that for any pair of fixed points there exists an infinite number of heteroclinic trajectories connecting them whereas the S corresponding set of heteroclinic points is dense in m . Exercise 1.13 Let g =={}n Î {}n = ,,¼ , - 1 S a : n Z S a : n 01 N1 1 and g =={}n Î {}n = ,,¼ , - 2 S b : n Z S b : n 01 N2 1 368 Homoclinic Chaos in Infinite-Dimensional Systems

C h be cycles (periodic trajectories). Prove that there exists a hetero- a g =g{}n Î clinic trajectory 12, S c : n Z that connects the cycles 1 p g t and 2 , i.e. such that e n n r dist()S c, g º infdS()cx, ® 0 , n ® -¥ 1 Î g x 1 6 and dist()Sn c, g = infdS()n cx, ® 0 ,.n ® +¥ 2 Î g x 2

For every NN there exists only a finite number of segments of the length . There- LL= { fore, the set of all segments is countable, i.e. we can assume that ak : =},,¼ k 12 , therewith the length of the segment ak +1 is not less than the length =S{}Î = of ak . Let us construct an element bbi : i Z from m taking bi 1 for £ i 0 and sequentially putting all the segments ak to the right of the zeroth position. As a result, we obtain an element of the form = ()¼,,,,,,,¼ Î L b 111a1 a2 a3 ,.aj (1.5)

g = {}n ³ Exercise 1.14 Prove that a positive semitrajectory + S b , n 0 with S ÎS b having the form (1.5) is dense in m , i.e. for every x m and e > 0 there exists nnx= (), e such that dxS()e, nb < . g Exercise 1.15 Prove that the semitrajectory + constructed in Exercise 1.14 e ÎS returns to an -vicinity of every point x m infinite number of times (Hint: see Exercises 1.4 and 1.9). g = {}n £ Exercise 1.16 Construct a negative semitrajectory – S c : n 0 S which is dense in m .

~~~~Thus, summing up the results of the exercises given above, we obtain the following assertion.~~~~

Theorem 1.1. ()S , The dynamical system m S of the Bernoulli shift of sequences ofm symbols possesses the properties: 1) there exists a finite number of fixed points; 2) there exist periodic orbits of any minimal period and the set of these S orbits is dense in the phase space m ; 3) the set of nonperiodic points is uncountable; 4) heteroclinic and homoclinic points are dense in the phase space; 5) there exist everywhere dense trajectories.

All these properties clearly imply the extraordinarity and complexity of the dyna- ()S , mics in the system m S . They also give a motivation for the following definitions. Exponential Dichotomy and Difference Equations 369

Let ()Xf, be a discrete dynamical system. The dynamics of the system ()Xf, is called chaotic if there exists a natural number kf such that the mapping k is topologically conjugate to the Bernoulli shift for some m , i.e. there exists a homeo- ® S ()k ()= ()() Î morphism h: X m such that hf x Shx for all xX . We also say that chaotic dynamics is observed in the system ()Xf, if there exist a number k and a set YX in invariant with respect to fk ()f kYYÌ such that the restriction of f k to Y is topologically conjugate to the Bernoulli shift. It turns out that if a dynamical system ()Xf, has a fixed point and a corresponding homoclinic trajectory, then chaotic dynamics can be observed in this system under some additional conditions (this assertion is the core of the Smale theorem). Therefore, we often speak about homoclinic chaos in this situation. It should also be noted that the approach presented here is just one of the possible methods used to describe chaotic behaviour (for example, other approaches can be found in [1] as well as in book [7], the latter contains a survey of methods used to study the dynamics of complicated systems and processes).

~~~~§ 2 Exponential Dichotomy and Difference Equations~~~~

This is an auxiliary section. Nonautonomous linear difference equations of the form = + Î xn +1 An xn hn ,,n Z (2.1) {} in a Banach space XA are considered here. We assume that n is a family of linear bounded operators in Xh , n is a sequence of vectors from X . Some results both ()º on the dichotomy (splitting) of solutions to homogeneous hn 0 equation (2.1) and on the existence and properties of bounded solutions to nonhomogeneous equation are given here. We mostly follow the arguments given in book [5] as well as in paper [4] devoted to the finite-dimensional case. {}Î Thus, let An : n Z be a sequence of linear bounded operators in a Banach space X . Let us consider a homogeneous difference equation = Î xn +1 An xn ,,nJ (2.2) where J is an interval in Z , i.e. a set of integers of the form = {}Î << JnZ : m1 nm2 , = -¥ = ¥ where m1 and m2 are given numbers, we allow the cases m1 and m2 + . {}Î Evidently, any solution xn : nJ to difference equation (2.2) possesses the property = $(), ³ , Î xm mnxn ,,mn mn J , 370 Homoclinic Chaos in Infinite-Dimensional Systems

C $(), = ××¼ >$(), = h where mn Am -1 An for mn and mm I . The mapping a $()mn, is called an evolutionary operator of problem (2.2). p t e Exercise 2.1 Prove that for all mnk³³ we have r $()$mk, = ()$mn, ()nk, . 6 {}Î 2 = Exercise 2.2 Let Pn : nJ be a family of projectors (i.e. Pn Pn ) in X = such that Pn +1 An An Pn . Show that $()$, = (), ³ , Î Pm mn mnPn ,,mn mn J , $(), i.e. the evolutionary operator mn maps Pn XP into m X . {} Exercise 2.3 Prove that solutions xn to nonhomogeneous difference equation (2.1) possess the property m - 1 = $(), + $(), + > xm mnxn å mk1 hk ,.mn kn=

{} Let us give the following definition. A family of linear bounded operators An is said to possess an exponential dichotomy over an interval J with constants > << {}Î K 0 and 0 q 1 if there exists a family of projectors Pn : nJ such that = , + Î a) Pn +1 An An Pn ,nn 1 J ; $(), £ mn- ³ , Î b)mnPn Kq ,mn ,mn J ; c) for nm³$ the evolutionary operator ()nm, is a one-to-one mapping ()- ()- of the subspace 1 Pm X onto 1 Pn X and the following estimate holds: $(), -1 - £ nm- £ , Î nm 1 Pn Kq ,,mn mn J . If these conditions are fulfilled, then it is also said that difference equation (2.2) admits an exponential dichotomy over JJ . It should be noted that the cases = Z and = J Z ± are the most interesting for further considerations, where Z+ Z– is the set of all nonnegative (nonpositive) integers. The simplest case when difference equation (2.2) admits an exponential dichotomy is described in the following example.

~~~~Example 2.1 (autonomous case)~~~~

Assume that equation (2.2) is autonomous, i.e. An º A for all n , and the spectrum s ()A does not intersect the unit circumference {}z Î C : z =1 . Linear operators possessing this property are often called hyperbolic (with respect to the fixed point x = 0 ). It is well-known (see, e.g., [8]) that in this case there exists a projector P with the properties: Exponential Dichotomy and Difference Equations 371

a)AP= PA , i.e. the subspaces PX and ()1-P X are invariant with respect to A ; s () b) the spectrum A PX of the restriction of the operator APX to lies strictly inside of the unit disc; s() c) the spectrum A ()1- P X of the restriction of A to the subspace ()1- P X lies outside the unit disc.

Exercise 2.4 Let CX be a linear bounded operator in a Banach space and let r º max {}z : z Î s()C be its spectral radius. Show that for > r ³ any q there exists a constant Mq 1 such that n £ n = ,,,¼ C Mq q , n 012 (Hint: use the formula r = lim Cn 1 n the proof of which can be n ® ¥ found in [9], for example).

~~~~Applying the result of Exercise 2.4 to the restriction of the operator A to PX , we obtain that there exist K > 0 and 0 < 0 and 0 <~~~~

~~~~Exercise 2.5 Assume that for the operator AP there exists a projector such that AP= PA and estimates (2.3) and (2.4) hold with 0 <~~~~

Thus, the hyperbolicity of the linear operator A is equivalent to the exponential di- = chotomy over Z of the difference equation xn +1 Axn with the projectors Pn independent of n . Therefore, the dichotomy property of difference equation (2.2) should be considered as an extension of the notion of hyperbolicity to the nonautonomous case. The meaning of this notion is explained in the following two exercises.

Exercise 2.6 Let A be a hyperbolic operator. Show that the space X can be decomposed into a direct sum of stable Xs and unstable Xu subspaces, i.e. XX= s + Xu therewith 372 Homoclinic Chaos in Infinite-Dimensional Systems

~~~~C n n s h A xKq£ x , xXÎ ,n ³ 0 , a p An xK³ -1 q-n x ,,,xXÎ u n ³ 0 t e with some constants K > 0 and 0 <~~~~

~~~~The next assertion (its proof can be found in the book [5]) plays an important role in the study of existence conditions of exponential dichotomy of a family of operators {}Î An : n Z .~~~~

Theorem 2.1. {}Î Let An : n Z be a sequence of linear bounded operators in a Ba- nach space X . Then the follfollooowingwing assertions are equivalent: {}Î (i) the sequence An : n Z possesses an exponential dichotomy over Z , (ii) for any bounded sequence {}hn : n Î Z from X there exists a unique {}Î bounded solution xn : n Z to the nonhomogeneous difference equation = + Î xn +1 An xn hn ,,.n Z . (2.5)

{} In the case when the sequence An possesses an exponential dichotomy, solutions to difference equation (2.5) can be constructed using the Green function: ì $(), ³ ï nmPm , nm, Gn(), m = í ï -[]$()mn, -1 ()1 - P , nm< . î m

Exercise 2.8 Prove that Gn(), m £ Kq nm- . {}Î Exercise 2.9 Prove that for any bounded sequence hn : n Z from X a solution to equation (2.5) has the form = (), + Î xn å Gn m 1 hm ,.n Z m Î Z Exponential Dichotomy and Difference Equations 373

~~~~Moreover, the following estimate is valid: + £ 1 q sup xn K sup hn . n 1- q n~~~~

The properties of the Green function enable us to prove the following assertion {} on the uniqueness of the family of projectors Pn . Lemma 2.1. {} Let a sequence An possess an exponential dichotomy over Z . Then {}Î the projectors Pn : n Z are uniquely defined.

Proof. {} {} Assume that there exist two collections of projectors Pn and Qn for {} (), which the sequence An possesses an exponential dichotomy. Let GP nm (), and GQ nm be Green functions constructed with the help of these collections. Then Theorem 2.1 enables us to state (see Exercise 2.9) that (), + = (), + å GP nm1 hm å GQ nm 1 hm m Î Z m Î Z Î {}Ì = for all n Z and for any bounded sequence hn X . Assuming that hm 0 ¹ - = = - for mk1 and hm h for mk1 , we find that (), = (), Î , Î ³ GP nkhGQ nkh ,,hX nk Z ,.nk = = This equality with nk gives us that Pn hQnh . Thus, the lemma is proved.

In particular, Theorem 2.1 implies that in order to prove the existence of an exponential dichotomy it is sufficient to make sure that equation (2.5) is uniquely solvable for any bounded right-hand side. It is convenient to consider this difference ¥ º ¥(), = {}Î equation in the space lX l Z X of sequences x xn: n Z of elements of X for which the norm ìü x ¥ =={}x sup íýx : n Î Z (2.6) l n l¥ îþn is finite. Assume that the condition ìü sup íýA : n Î Z < ¥ (2.7) îþn = {}Î ¥ {}= - ¥ is valid. Then for any x xn lX the sequence yn xn An -1 xn -1 lies in lX . Consequently, equation ()= - = {}Î ¥ L x n xn An -1 xn -1 ,(2.8)x xn lX ¥ = ¥(), defines a linear bounded operator acting in the space lX l Z X . Therewith assertion (ii) of Theorem 2.1 is equivalent to the assertion on the invertibility of the operator L given by equation (2.8). 374 Homoclinic Chaos in Infinite-Dimensional Systems

C h The assertion given below provides a sufficient condition of invertibility of the a operator L . Due to Theorem 2.1 this condition guarantees the existence of an expo- p t nential dichotomy for the corresponding difference equation. This assertion will be e used in Section 4 in the proof of Anosov’s lemma. It is a slightly weakened variant r of a lemma proved in [6]. 6 Theorem 2.2. {}Î Assume that a sequence of operators An: n Z satisfies condition {}Î (2.7). Let there exist a family of projectors Qn : n Z such that £ - £ Qn K ,,,1 Qn K , (2.9) ()d- £ ()- £ d Qn +1 An 1 Qn ,,,1 Qn +1 An Qn , (2.10) Î ()- for all n Z . We also assume that the operator 1 Qn +1 An is invertible ()- ()- as a mapping from 1 Qn X into 1 Qn +1 X and the estimates £ l []()- -1 ()l- £ AnQn , 1 Qn +1 An 1 Qn +1 (2.11) are valid for every n Î Z . If K l £d1 ,,,£ 1 , (2.12) 8 8 ¥ then the operator Ll acting in X according to formula (2.8) is invertible and L-1 £ 2 K +1 .

Proof. Let us first prove the injectivity of the mapping L . Assume that there exists a = {}Î ¥ = = Î nonzero element x xn lX such that Lx 0 , i.e. xn An -1 xn -1 for all n Z . {} Let us prove that the sequence xn possesses the property ()- £ 1 Qn xn Qn xn (2.13) for all n Î Z . Indeed, let there exist m Î Z such that ()- > 1 Qm xm Qm xm .(2.14) ()- > It is evident that this equation is only possible when 1 Qm xm 0 . Let us consider the value º ()- - = Nm +1 1 Qm +1 xm +1 Qm +1 xm +1 = ()- - 1 Qm +1 Am xm Qm +1 Am xm . (2.15) It is clear that ()- ³ ()- ()- - ()- 1 Qn +1 An xn 1 Qn +1 An 1 Qn xn 1 Qn +1 An Qn xn . Since []()- -1 ()- ()- = ()- 1 Qn +1 An 1 Qn +1 An 1 Qn 1 Qn , Exponential Dichotomy and Difference Equations 375 it follows from (2.11) that ()- £ l ()- ()- 1 Qn x 1 Qn +1 An 1 Qn x for every xXÎ and for all n Î Z . Therefore, we use estimates (2.10) to find that ()- ³ l-1 ()- - d 1 Qn +1 An xn 1 Qn xn xn .(2.16) Then it is evident that ££+ ()- Qn +1 An xn Qn +1 An Qn xn Qn +1 An 1 Qn xn

£ æöQ × A Q + Q A ()1- Q x . èøn +1 n n n +1 n n n Therefore, estimates (2.9)–(2.11) imply that £ ()ld+ Î Qn +1 An xn K xn ,.n Z (2.17) Thus, equations (2.15)–(2.17) lead us to the estimate ³ l-1 ()- - ()d + l Nm +1 1 Qm xm 2 K xm . It follows from (2.14) that £ + ()- < ()- xm Qm xm 1 Qm xm 21Qm xm . Therefore, > ()l-1 - l - d ()- Nm +1 2K 4 1 Qm xm . Hence, if conditions (2.12) hold, then ()- - >>()- 1 Qm +1 xm +1 Qm +1 xm +1 71Qm xm 0 .(2.18) When proving (2.18) we use the fact that l-1 ³³ ³ 8K 8 Qn 8 . > > Thus, equation (2.18) follows from (2.14), i.e. Nm 0 implies Nm +1 0 . Hence, ()- > ³ 1 Qn xn Qn xn for allnm . Moreover, (2.18) gives us that × ³³()- nm- ()- ³ Kxn 1 Qn xn 7 1 Qm xm ,.nm ® ¥ ® ¥ = {}Î ¥ Therefore, xn + as n + . This contradicts the assumption x xn lX . Thus, for all n Î Z estimate (2.13) is valid. In particular it leads us to the inequality ££()- + xn 1 Qn xn Qn xn 2 Qn xn .(2.19) Therefore, it follows from (2.17) that = £ ()ld+ Qn +1 xn +1 Qn +1 An xn 2 K Qn xn for all n Î Z . We use conditions (2.12) to find that Q x £ 1 Q x ,.n Î Z (2.20) n +1 n +1 2 n n 376 Homoclinic Chaos in Infinite-Dimensional Systems

C = {}¹ Î h If x xn 0 , then inequality (2.19) gives us that there exists m Z such that a Q x ¹ 0 . Therefore, it follows from (2.20) that p m m t ³ mn- > e Qn xn 2 Qmxm 0 r £ ® -¥ ® ¥ for all nm . We tend n to obtain that Qnxn + which is impossible 6 {} due to (2.9) and the boundedness of the sequence xn . Therefore, there does not Î ¥ = exist a nonzero x lX such that L x 0 . Thus, the mapping L is injective. Let us now prove the surjectivity of LR . Let us consider an operator in the ¥ space lX acting according to the formula ()= - ()- = {}Î ¥ R y n Qn yn Bn 1 Qn +1 yn +1 ,,y yn lX = []()- -1 ()- ()- where the operator Bn 1 Qn +1 An acts from 1 Qn +1 X into 1 Qn X and is inverse to ()1- Q A . It follows from (2.9) and (2.11) that n +1 n ()- 1 Qn X ¥ Ry £ ()K + l y ,.y Î l (2.21) l ¥ l ¥ X It is evident that ()- = -()- - ()- - LRy n yn 1 Qn yn Bn 1 Qn +1 yn +1 - + ()- An -1 Qn -1 yn -1 An -1 Bn -1 1 Qn yn . Since ()- - = - 1 Qn An -1 Bn -1 1 Qn 1 Qn , we have that ()- = - ()- - + LRy n yn Bn 1 Qn +1 yn +1 An -1 Qn -1 yn -1 + ()- ()- Qn An -1 1 Qn -1 Bn -1 1 Qn yn . Consequently, ()- £ ()- × ++× LRy n yn Bn 1 Qn +1 yn +1 An -1 Qn -1 yn -1 + ()- ××()- Qn An -1 1 Qn -1 Bn -1 1 Qn yn . Therefore, inequalities (2.10), (2.11), and (2.12) give us that 1 LRyy- ££l ()2 + d y y , l ¥ l ¥ 2 l ¥ i.e. LR -1 £ 12¤ . That means that the operator LR is invertible and - - ()LR 1 ££()1- LR- 1 1 2 .(2.22) = {} ¥ = Let h hn be an arbitrary element of lX . Then it is evident that the element y = RLR()-1 h is a solution to equation Lyh= . Moreover, it follows from (2.21) and (2.22) that y £ 2 ()K +l h . l ¥ l ¥ Hence, LL is surjective and -1 £ 2 K +1 . Theorem 2.2 is proved. Hyperbolicity of Invariant Sets for Differentiable Mappings 377

~~~~§ 3 Hyperbolicity of Invariant Sets for Differentiable Mappings~~~~

Let us remind the definition of the differentiable mapping. Let XY and be Banach spaces and let U be an open set in Xf . The mapping from U into Y is called (Frechét) differentiable at the point x Î U if there exists a linear bounded operator Df() x from XY into such that 1 lim fx()+ v - fx()- Df() x v = 0 . v ® 0 v If the mapping fx is differentiable at every point ÎU , then the mapping Df: xDfx® () acts from UL into the Banach space ()XY, of all linear bounded operators from XYDf into . If : UL® ()XY, is continuous, then the mapping f is said to be continuously differentiable (or C1 -mapping) on U . The notion of the derivative of any order can also be introduced by means of induction. For example, D2 fx() is the Frechét derivative of the mapping Df: UL® ()XY, .

Exercise 3.1 Let gf and be continuously differentiable mappings from U Ì X into Y and from W Ì Y into Z , respectively. Moreover, let UW and be open sets such that g()U Ì W . Prove that ()f ° g ()=x = fgx()() is a C1 -mapping on U and obtain a chain rule for the differentiation of a composed function Df()° g ()x = Df() g() x Dg() x ,.x Î U

Exercise 3.2 Let fX be a continuously differentiable mapping from into Xf and let n be the nff -th degree of the mapping , i.e. n()=x = ff()n -1()x , n ³ 1 ,f 1()x º fx() . Prove that f n is a C1 -mapping on X and ()Df n ()x = Df() fn -1()x ×××Df()¼ fn -2()x Df() x . (3.1)

Now we give the definition of a hyperbolic set. Assume that f is a continuously differentiable mapping from a Banach space X into itself and L is a subset in X which is invariant with respect to ff ()L()L Ì L . The set is called hyperbolic (with respect to fPx ) if there exists a collection of projectors {}(): x ÎL such that a)Px() continuously depends on x ÎL with respect to the operator norm; b) for every x Î L Df() x × Px()= Pfx()()× Df() x ;(3.2) c) the mappings Df() x are invertible for every x Î L as linear operators from ()1- Px()XIPfx into ()- ()() X ; 378 Homoclinic Chaos in Infinite-Dimensional Systems

~~C Î L h d) for every x the following equations hold: a n n p ()Df ()x Px() £ Kq ,,n ³ 0 (3.3) t - e []()Dfn ()x 1 ()1- Pf()n ()x £ Kqn ,,n ³ 0 (3.4) r 6 with the constants K > 0 and 0 <~~

L = {} 1 Exercise 3.3 Let x0 , where x0 is a fixed point of a C -mapping f , ()=L i.e. fx0 x0 . Then for the set to be hyperbolic it is necessary () and sufficient that the spectrum of the linear operator Df x0 does not intersect the unit circumference (Hint: see Example 2.1).

L 1 g = {}Î Let be an invariant hyperbolic set of a C -mapping f and let xn : n Z L g =L{} be a complete trajectory (in ) for f , i.e. xn is a sequence of points from ()= Î such that fxn xn +1 for all n Z . Let us consider a difference equation obtained as a result of linearization of the mapping f along g : = () Î un +1 Df xn un ,.n Z (3.5)

~~~~Exercise 3.4 Prove that the evolutionary operator $()mn, of difference equation (3.5) has the form $(), = ()mn- () > , Î mn Df xn ,,mn mn Z .~~~~

Exercise 3.5 Prove that difference equation (3.5) admits an exponential dichotomy over Z with (i) the constants Kq and given by equations = () (3.3) and (3.4) and (ii) the projectors Pn Pxn involved in the definition of the hyperbolicity.

It should be noted that property (a) of uniform continuity implies that the projectors Px() are similar to one another, provided the values of x are close enough. The proof of this fact is based on the following assertion.

~~~~Lemma 3.1. Let PQ and be projectors in a Banach space X . Assume that~~~~

~~~~PK£ ,,1- P £ K PQ- < 1 (3.6) 2K for some constant K ³ 1 . Then the operator JPQ= + ()1- P 1- Q (3.7) Hyperbolicity of Invariant Sets for Differentiable Mappings 379~~~~

~~~~possesses the property PJ= JQ and is invertible, therewith - J -1 £ ()12- KPQ× - 1 . (3.8)~~~~

Proof. Since P2 + ()1- P 2 = 1 , we have J -1 ==JP- 2 - ()1- P 2 PQ()- P + ()1- P ()PQ- . It follows from (3.6) that J - 1 £ 2 KP- Q < 1 . Hence, the operator J-1 can be defined as the following absolutely convergent series ¥ J-1 = å ()1- J n . n = 0 This implies estimate (3.8). The permutability property PJ= JQ is evident. Lemma 3.1 is proved.

Exercise 3.6 Let L be a connected compact set and let {}Px(): x Î L be a family of projectors for which condition (a) of the hyperbolicity definition holds. Then all operators Px() are similar to one another, i.e. ,LÎ = for any xy there exists an invertible operator JJxy, such that Px()= JP() y J-1 .

The following assertion contains a description of a situation when the hyperbolicity of the invariant set is equivalent to the existence of an exponential dichotomy for difference equation (3.5) (cf. Exercise 3.5).

Theorem 3.1. Let fx() be a continuously differentiable mapping of the space X into ()= {}Î itself. Let x0 be a hyperbolic fixed point of ffx (0 x0 ) and let yn: n Z be a homoclinic trajectory (not equal to x0 ) of the mapping f , i.e. ()= Î ® ® ±¥ fyn yn +1 ,,,n Z ,,,yn x0 ,,.n . (3.9) L = {}È {}Î Then the set x0 yn : n Z is hyperbolic if and only if the difference equation = () Î un +1 Df yn un ,,,n Z , (3.10) possesses an exponential dichotomy over Z .

Proof. If L is hyperbolic, then (see Exercise 3.5) equation (3.10) possesses an exponential dichotomy over Z . Let us prove the converse assertion. Assume that equa- {}Î tion (3.10) possesses an exponential dichotomy over Z with projectors Pn: n Z 380 Homoclinic Chaos in Infinite-Dimensional Systems

C () h and constants Kq and . Let us denote the spectral projector of the operator Dfx0 a corresponding to the part of the spectrum inside the unit disc by P . Without loss p t of generality we can assume that e []()n £ n ³ r Df x0 PKq,,n 0 6 []()-n ()- £ -n ³ Df x0 1 P Kq ,.n 0 Î L () ()= ()= Thus, for every x the projector Px is defined: Px0 P , Pyk Pk . The structure of the evolutionary operator of difference equation (3.10) (see Exer- cise 3.4) enables us to verify properties (b)–(d) of the definition of a hyperbolic set. In order to prove property (a) it is sufficient to verify that - ® ® ±¥ Pk P 0 ask . (3.11) Since L is a compact set, then MDfx= sup {}(): x Î L < ¥ .(3.12) Let us consider the following difference equations = () Î vn +1 Df x0 vn ,,n Z (3.13) and ()k = ()()k Î wn +1 Df ynk+ wn ,,n Z (3.14) where k is an integer. It is evident that equation (3.14) admits an exponential di- ()k = (), chotomy over Z with constants Kq and and projectors Pn Pnk+ . Let Gn m and G()k ()nm, be the Green functions (see Section 2) of difference equations (3.13) and (3.14). We consider the sequence = (), - ()k (), Î xn Gn0 zG n 0 z ,.zX Since (see Exercise 2.8) Gn(), 0 £ Kqn ,,(3.15)G()k ()n, 0 £ Kq n {} we have that the sequence xn is bounded. Moreover, it is easy to prove (see Exer- {} cise 2.9) that xn is a solution to the difference equation - () = º []()- ()()k (), xn +1 Df x0 xn hn Df x0 Df ynk+ G n 0 z . {} It follows from (3.12) and (3.15) that the sequence hn is bounded. Therefore, (see Exercise 2.9), (), - ()k (), º = (), + Gn 0 zG n 0 z xn å Gn m 1 hm . m Î Z If we take n = 0 in this formula, then from the definition of the Green function we obtain that ()- = (), + ()()- ()()k (), PPk zGå 0 m 1 Df x0 Df ymk+ G m 0 z . m Î Z Hyperbolicity of InvariantAnosov’s Sets for Lemma Differentiable on e -trajectories Mappings 381

~~~~Therefore, equation (3.15) implies that ()- £ 2 ()- ()××mm+ +1 PPk zKå Df x0 Df ymk+ q z . m Î Z Consequently,~~~~

- ££2 ()- ()× 2 m -1 PPk K å Df x0 Df ymk+ q m Î Z ìü £ 2 ()- ()× 2 m -1 + 2 m -1 K íýmax Df x0 Df ymk+ å q 2 Mqå , îþmN£ mN£ mN> where N is an arbitrary natural number. Upon simple calculations we find that 2 - £ 2 K æö()- ()+ 2 N +2 PPk max Df x0 Df ymk+ 2 Mq q ()1- q2 èømN£ for every N ³ 1 . It follows that 2 - £ 4 K M 2 N +1 = ,,¼ lim PPk q ,.N 12 k ® ±¥ 1- q2 We assume that N ® + ¥ to obtain that - £ limPPk 0 . k ® ±¥ This implies equation (3.11). Therefore, Theorem 3.1 is proved.

L = {}È {}Î It should be noted that in the case when the set x0 yn: n Z from The- g = {}Î orem 3.1 is hyperbolic the elements yn of the homoclinic trajectory yn: n Z are called transversal homoclinic points. The point is that in some cases (see, e.g., [4]) it can be proved that the hyperbolicity of L is equivalent to the transversa- s () u () lity property at every point yn of the stable W x0 and unstable W x0 manifolds of a fixed point x0 (roughly speaking, transversality means that the surfaces s () u() W x0 and W x0 intersect at the point yn at a nonzero angle). In this case the trajectory g is often called a transversal homoclinic trajectory.

~~~~§ 4 Anosov’s Lemma on e -trajectories~~~~

1 {}Î Let fC be a -mapping of a Banach space Xy into itself. A sequence n : n Z inX is called a dd-pseudotrajectory (or -pseudoorbit) of the mapping f if for all n Î Z the equation -d()£ yn +1 fyn 382 Homoclinic Chaos in Infinite-Dimensional Systems

C {}Î e-trajectory h is valid. A sequence xn: n Z is called an -trajectory of the mapping f cor- a responding to a d -pseudotrajectory {}y : n Î Z if p n ()= Î t (a)fxn xn +1 for any n Z ; e r -e£ Î (b)xn yn for all n Z . {} 6 It should be noted that condition (a) means that xn is an orbit (complete trajectory) of the mapping fC . Moreover, if a pair of 1 -mappings fg and is given, then the notion of the e -trajectory of the mapping g corresponding to a d -pseudoorbit of the mapping f can be introduced. The following assertion is the main result of this section.

Theorem 4.1. Let fC be a 1 -mapping of a Banach space X into itself and let L be a hyperbolic invariant ()f ()L Ì L set. Assume that there exists a D -vicinity O of the set L such that fx() and Df() x are bounded and uniformly O O e > continuous on the closure of the set . Then there exists 0 0 possessing the property that for every 0 0 there exists 0 such d {}Î L e that any -pseudoorbit yn: n Z lying in has a unique -trajectory { Î} {} xn: n Z corresponding to yn .

~~~~As the following theorem shows, the property of the mapping f to have an e -trajectory is rough, i.e. this property also remains true for mappings that are close to f .~~~~

Theorem 4.2. Assume that the hypotheses of Theorem 4.1 hold for the mapping f . Let Wh ()f be a set of continuously differentiable mappings g of the space X into itself such that the following estimates hold on the closure O of the D -vicinityO of the set L : fx()-hgx() < ,,.Df() x -hDg() x < . (4.1) e >eÎ ( , e ] Then 0 0 can be chosen to possess the property that for every 0 0 there exist de=hhed()> 0 and =d()> 0 such that for any -pseudotra- {}Î L ÎW () jectory yn: n Z (lying in ) of the mapping fg and for any h f {}Î there exists a unique trajectory xn : n Z of the mapping g with the property -e£ Î yn xn for all n Z .

It is clear that Theorem 4.1 is a corollary of Theorem 4.2 the proof of which is based on the lemmata below. Hyperbolicity of InvariantAnosov’s Sets for Lemma Differentiable on e -trajectories Mappings 383

Lemma 4.1. Let U be an open set in a Banach space X and let : U ® X be a continuously differentiable mapping. Assume that for some point y Î U [] ()-1 e > there exist an operator D y and a number 0 0 such that -1 ()- () £ æö[] ()-1 D x D y èø2 D y (4.2) -e£ e Î( , e ] for all xxy with the property 0 . Assume that for some 0 0 the inequality -1 () £ × e æö[] ()-1 y q èø2 D y (4.3) is valid with 0 <

~~~~Proof. - - Let G = ()2 []D ()y 1 1 and let h()x = ()x - ()y - D ()y ()xy- .~~~~

For x , x Î Be º {}z : zy-e£ we have that 1 2 0 0 h()h- ()== ()- ()- ()()- x1 x2 x1 x2 D y x1 x2 1 = () ()+ x()- - ()()x- ò D x1 x2 x1 D y x1 x2 d . 0 Since 1 h()h- ()£ ()+ x ()- -x () - x1 x2 ò D x1 x2 x1 D y d x1 x2 , 0 it follows from (4.2) that h()h-G()£ - x1 x2 x1 x2 (4.6) for all x and x from Be . Now we rewrite the equation G()x = 0 in the form 1 2 0 - xTx= ()º yD- [] ()y 1 ()G()x - ()x ++ ()y h()x . 384 Homoclinic Chaos in Infinite-Dimensional Systems

C = { h Let us show that the mapping TB has a unique fixed point in the ball e x : a xy-e£}. It is evident that p t ()- £ [] ()-1 æöG()- ()++ () h() e Tx y D y èøx x y x r Îh()= 6 for any xBe . Since y 0 , we obtain from (4.6) that h()x = h()x -G h()y ££xy-Ge. Therefore, estimates (4.3) and (4.4) imply that

~~~~Tx()-ey £ forxBÎ e ,~~~~

i.e. TB maps the ball e into itself. This mapping is contractive in Be . Indeed, æö Tx()- Tx()£ 1 H ()hx , x + ()hx - ()x , 1 2 2G èø1 2 1 2 where H (), º G ()- G()- ()+ ()= x1 x2 x1 x2 x1 x2 1 = []xG ()+ x ()- - ()+ x()- ()- ò D x1 x1 x2 D x1 x1 x2 d x1 x2 . 0 It follows from (4.5) that H ()x , x £ 1 G x - x . 1 2 2 1 2 This equation and inequality (4.6) imply the estimate

~~~~Tx()- Tx()£ 3 x - x . 1 2 4 1 2~~~~

~~~~Therefore, the mapping TB has a unique fixed point in the ball e = {x : xy-e£}. The lemma is proved.~~~~

Let the hypotheses of Theorems 4.1 and 4.2 hold. We assume that h < 1 in (4.1). Then for any element g ÎWh()f the following estimates hold: gx() £ M ,,,(4.7)Dg() x £ M x Î O where M > 0 is a constant. In particular, these estimates are valid for the mapping f .

Lemma 4.2. {}dÎ L Let yn: n Z be a -pseudotrajectory of the mapping f lying in . ³ {}dº Î × Then for any k 1 the sequence zn ynk: n Z is a Mk -1 -pseu- k dotrajectory of the mapping f . Here Mk has the form = ++¼ +k ³ = Mk 1 M M ,,,k 1 M0 1 (4.8) and M is a constant from (4.7). Hyperbolicity of InvariantAnosov’s Sets for Lemma Differentiable on e -trajectories Mappings 385

Proof. Let us use induction to prove that -di ()£ × ££ ynk+ i f ynk Mi -1 ,.1 ik (4.9) {} d = Since yn is a -pseudotrajectory, then it is evident that for i 1 inequality (4.9) is valid. Assume that equation (4.9) is valid for some i ³ 1 and prove estimate (4.9) for i + 1 : - i +1()£ - ()+ ()- ()i() ynk++ i 1 f ynk ynk++ i 1 fynk+ i fynk+ i ff ynk . With the help of (4.7) we obtain that -di +1()££d+d- i() + × d = × ynk++ i 1 f ynk Mynk+ i f ynk MMi-1 Mi . Thus, Lemma 4.2 is proved.

Lemma 4.3. {} d L {} Let yn be a -pseudoorbit of the mapping f lying in . Let xn be a trajectory of the mapping g Î Wh ()f such that -e£ Î ynk xnk ,,n Z (4.10) for some k ³ 1 . If ()ed, + h £ D max Mk , (4.11) then -edh£ (), + × yn xn max Mk , (4.12) where Mk has the form (4.8).

Proof. We first note that - = - ()£ ynk+1 xnk+1 ynk+1 gxnk £ - ()++()- () ()- () ynk+1 fynk fynk gynk gynk gxnk . Therefore, it is evident that -dh££++- () edh, + ()+ ynk+1 xnk+1 Mynk xnk max 1 M .(4.13) Here we use the estimate 1 gy()- gx() ££ò Dg()x y + x ()xy- d × xy- Mx- y 0 which follows from (4.7) and holds when the segment connecting the points x and y lies in O . Condition (4.11) guarantees the fulfillment of this property at each stage of reasoning. If we repeat the arguments from the proof of (4.13), then it is easy to complete the proof of (4.12) using induction as in Lemma 4.2. Lemma 4.3 is proved. 386 Homoclinic Chaos in Infinite-Dimensional Systems

~~~~C h Lemma 4.4. a ÎL -e£ p Let y and xy . Assume that t k -1 e max()eh, ()D 1 ++¼ M < . (4.14) r Then the estimates 6 f j()y - f j()x £ M j xy- ,,Dfj()x £ M j (4.15)~~~~

~~~~f j()y -ehg j()x £ max(), ()1 ++¼ M j , (4.16)~~~~

~~~~f j()x -hg j()x £ ()1 ++¼ M j -1 (4.17)~~~~

~~~~are valid for j = 12,,¼ ,k and for every mapping g Î Wh ()f .~~~~

Proof. As above, let us use induction. If j = 1 , it is evident that equations (4.15)– (4.17) hold. The transition from jj to +1 in (4.15) is evident. Let us consider estimate (4.16): j +1()- j +1()= æö()j()- ()j() + ()j()- ()j() f y g x èøff y fg x fg x gg x .(4.18) Condition (4.14) and the induction assumption give us that gj ()x lies in the ball with the centre at the point f j ()y Î L lying in O . Therefore, it follows from (4.18) that f j +1()y - g j +1()x ££Mfj()y -hg j ()x +ehmax(), () 1 ++¼ M j +1 . The transition from j to j +1 in (4.17) can be made in a similar way. Lemma 4.4 is proved.

Lemma 4.5. There exists D¢ £ D such that the equations ìü sup íýf k()x - g k()x : x ÎO ¢ £ r ()h (4.19) îþk and ìü sup íý()Dfk ()x - ()Dgk ()x : x Î O ¢ £ r ()h (4.20) îþk are valid in the D¢ -vicinity O ¢ of the set L for any function ÎrW () ()h ®h® g h f . Here k 0 as 0 .

The proof follows from the definition of the class of functions Wh ()f and estimates (4.7) and (4.17). Hyperbolicity of InvariantAnosov’s Sets for Lemma Differentiable on e -trajectories Mappings 387

Let us also introduce the values ìü w () = sup íýDf k()y - Dfk()x : y Î L , xy-£ (4.21) k îþ and ìü w()= sup íýPy()- Px(), xy, Î L , xy-£ .(4.22) îþ The requirement of the uniform continuity of the derivative Df() x (see the hypotheses of Theorem 4.1) and the projectors Px() (see the hyperbolicity definition) enables us to state that w () ®w()®® k 0 ,as.(4.23)0 0 {} d L Let yn be a -pseudotrajectory of the mapping f lying in . Then due to {}= Î d Lemma 4.2 the sequence yn ynk: n Z is a Mk -1 -pseudotrajectory of the mapping f k . Let us consider the mappings ()x and G()x in the space ¥ º ¥(), lX l Z X (for the definition see Section 2) given by the equalities [] () = + - k()+ x n yn xn f yn -1 xn -1 ,(4.24) []G() = + - k()+ x n yn xn g yn -1 xn -1 , (4.25) = {}Î ¥ e where x xn : n Z is an element from lX . Thus, the construction of -trajec- k k {} tories of the mapping f and g corresponding to the sequence yn is reduced to solving of the equations ()x = 0 and G()x = 0 in the ball {}x: x ¥ £ e . Let us show that for k large enough Lemma 4.1 can be lX applied to the mappings and G . Let us start with the mapping .

~~~~Lemma 4.6. 1 ¥ The function is a C -smooth mapping in lX with the properties () £ d 0 Mk -1 , (4.26) ()-w () £ ()e £ e D x D 0 k ,.x ¥ (4.27) lX~~~~

Proof. {} d Estimate (4.26) follows from the fact that yn is a Mk -1 -pseudotrajectory. Then it is evident that [] () = - k()+ D x h n hn Df yn -1 xn -1 hn -1 ,(4.28) = {}Î = {}Î ¥ where x xn : n Z and h hn: n Z lie in lX . Therefore, simple calculations and equation (4.21) give us (4.27). 388 Homoclinic Chaos in Infinite-Dimensional Systems

C h In order to deduce relations (4.2) and (4.3) from inequalities (4.26) and (4.27) for a y = 0 , we use Theorem 2.2. Consider the operator LD= ()0 . It is clear that p t []= - k () = {} e L h n hn Df yn -1 hn -1 ,.h hn r = k() = Let us show that equations (2.9)–(2.11) are valid for An Df yn -1 and Qn 6 = () Pyn -1 and then estimate the corresponding constants. Property (2.9) follows from the hyperbolicity definition. Equations (3.3) and (4.15) imply that £ k £ k An Qn Kq andAn M . Further, the permutability property (3.2) gives us that ()- ==[]- ()- Qn +1 An 1 Qn Qn +1 An An Qn 1 Qn

= - ()k()()- Qn +1 Pf yn -1 An 1 Qn . Hence (see (4.22)), ()w- ££()d ××- w()d k × Qn +1 An 1 Qn Mk -1 An 1 Qn Mk -1 M K . Similarly, we find that ()- £ w()d k × 1 Qn +1 AnQn Mk -1 M K . The operator = ()k()- ()- ()- ()k() Jn Qn +1 Pf yn -1 1 Qn +1 1 Pf yn -1 is invertible if (see Lemma 3.1) Q -wPf()k()y £ ()dM < 1 . n +1 n -1 k -1 2K Moreover, -1 £ ()- w()d -1 < Jn 12K Mk -1 2 , w()d 0 . If Hyperbolicity of InvariantAnosov’s Sets for Lemma Differentiable on e -trajectories Mappings 389

~~~~- - dM £ 1 ()4K +2 1 × e ,,(4.30)w ()e £ ()4K + 2 1 k -1 2 k then by Lemma 4.6 relations (4.2) and (4.3) hold with y =ee0 , = , and q = 12¤ . If~~~~

r ()h £ 1 min()e, 1 () 4K + 2 -1 ,(4.31) k 2 then equations (4.4) and (4.5) also hold with y =ee0 , = , and q = 12¤ . Hence, under conditions (4.29)–(4.31) there exists a unique solution to equation G()x = 0 possessing the property £ e . This means that for any d -pseudoorbit {y : x l ¥ n Î} L {}Î n Z (lying in ) of the mapping fz there exists a unique trajectory n: n Z of the mapping g Î Wh ()f such that -e£ Î znk ynk ,,n Z provided conditions (4.29)–(4.31) hold. Therefore, under the additional condition ()edh++ £ D Mk and due to Lemma 4.3 we get -edh£ ()++ Î yn zn Mk ,.n Z These properties are sufficient for the completion of the proof of Theorem 4.2. 3 k £e££D¢ D D¢ Let us fix k such that 16 K q 1 . We choose 0 ( is defined in Lemma 4.5) such that e w ()e £ ()+ -1 e £ 0 k 4K 2 for all . 2Mk e Îee( , e ] = []-1 Let us fix an arbitrary 0 0 and take 2 Mk . Now we choose de=hhed() and = () such that the following conditions hold: w()d < kw()d £ 4 K Mk -1 1 ,,8KM Mk -1 1 - - dM £ 1 e ()2K + 1 1 ,,.r ()h £ 1 min()e, 1 () 2K + 1 1 2 ()dh+ M £ e k -1 4 k 4 k It is clear that under such a choice of dhd and any -pseudoorbit (from L ) of the mapping f has a unique e -trajectory of the mapping g . Thus, Theorem 4.2 is proved.

Exercise 4.1 Let the hypotheses of Theorem 4.1 hold. Show that there D >d> {}Î exist0 and 0 such that for any two trajectories xn: n Z {}Î (), and yn : n Z of a dynamical system Xf the conditions dist()x , L < D ,,dist()y , L < D sup x - y £ d n n n n n º Î imply that xn yn , n Z . In other words, any two trajectories of the system ()Xf, that are close to a hyperbolic invariant set cannot remain arbitrarily close to each other all the time. 390 Homoclinic Chaos in Infinite-Dimensional Systems

C h Exercise 4.2 Show that Theorem 4.1 admits the following strengthening: a e > p if the hypotheses of Theorem 4.1 hold, then there exists 0 0 such t e Îdde(), e = () that for every 0 0 there exists with the property e d {} (), L < d r that for any -pseudoorbit yn such that dist yn there e 6 exists a unique -trajectory. Exercise 4.3 Prove the analogue of the assertion of Exercise 4.2 for Theo- rem 4.2. L = {}Î Exercise 4.4 Let xn : n Z be a periodic orbit of the mapping f , k()º = Î ³ i.e. f xn xnk+ xn for all n Z and for some k 1 . Assume that the hypotheses of Theorem 4.2 hold. Then for h > 0 small enough every mapping g Î Wh ()f possesses a periodic trajectory of the period k .

~~~~§ 5 Birkhoff-Smale Theorem~~~~

One of the most interesting corollaries of Anosov’s lemma is the Birkhoff-Smale theorem that provides conditions under which the chaotic dynamics is observed in a discrete dynamical system ()Xf, . We remind (see Section 1) that by definition the possibility of chaotic dynamics means that there exists an invariant set Y in the space Xf such that the restriction of some degree k of the mapping fY on is to- S pologically equivalent to the Bernoulli shift S in the space m of two-sided infinite sequences of m symbols.

Theorem 5.1. Let fX be a continuously differentiable mapping of a Banach space Î {}Î into itself. Let x0 X be a hyperbolic fixed point of fy and let n: n Z be a homoclinic trajectory of the mapping f that does not coincide with

x0 , i.e. ()= ()= ¹ Î ® ® ±¥ fx0 x0 ;;,fyn yn +1 ,,,yn x0 ,,;n Z ; yn x0 , n . {}Î Assume that the trajectory yn : n Z is transversal, i.e. the set L = {}È {}Î x0 yn : n Z is hyperbolic with respect to f and there exists a vicinity O of the set L such that fx() and Df() x are bounded and uniformly continuous on the closure O . By Wh()f we denote a set of continuously differentiable mappings gX of the space into itself such that fx()-hgx()£ ,,,Df() x -hDg() x £ ,,.x ÎO . Birkhoff-Smale Theorem 391

Then there exists h > 0 such that for any mapping g Î Wh()f and for any m ³ 2 there exist a natural number l and a continuous mapping j of the S º jS() space m into a compact subset Y m in X such that = jS() l l ()= a) Y m is strictly invariant with respect to g , i.e. g Y Y ; = ()¼ ,,,¼ ¢¼= ()¢ ,,,¢ ¢ ¼ b) if a a-1 a0 a1 and a a-1 a0 a1 are elements S ¹ ¢ ³j()¹ j()¢ of m such that ai ai for some i 0 , then a a ; c) the restriction of g l on Y is topologically conjugate to the Bernoulli S shift S in m , i.e. l()jj()= () Î S g a Sa ,,.a m . Moreover, if in addition we assume that for the mapping g there exists e > {}Î {}Î 0 0 such that for any two trajectories xn : n Z and xn : n Z e L (of the mapping g ) lying in the 0 -vicinity of the set the condition x = x for some n Î Z implies that x = x for all n Î Z , then the map- n0 n0 0 n n ping j is a homeomorphism.

The proof of this theorem is based on Anosov’s lemma and mostly follows the standard scheme (see, e.g., [4]) used in the finite-dimensional case. The only difficulty arising in the infinite-dimensional case is the proof of the continuity of the mapping j. It can be overcome with the help of the lemma presented below which is bor- rowed from the thesis by Jürgen Kalkbrenner (Augsburg, 1994) in fact. It should also be noted that the condition under which j is a homeomorphism holds if the mapping g does not “glue” the points in some vicinity of the set L , i.e. the equality gx()= gx() implies xx= .

Lemma 5.1. Let the hypotheses of Theorem 5.1 hold. Let us introduce the notation º (), m = {}Î -mn£ Jn Jn k0 k Z : kk0 , Îmn, Î = {}Î where k0 Z and N . Let zzn: nJn be a segment (lying in Ld ) of a- pseudoorbit of the mapping f : Î L -d()£ , + Î zn ,,.zn +1 fzn nn 1 Jn (5.1) = {}Î = {}Î Assume that xxn : nJn and xxn : nJn are segments of orbits of the mapping g Î Wh()f : ()= ()= , + Î gxn xn +1 ,,,gxn xn +1 nn 1 Jn (5.2) such that -e£ -e£ zn xn ,.zn xn (5.3) Then there exist dhe,, >m0 , and Î N such that conditions (5.1)– (5.3) imply the inequality -n x - x £ 21 e . (5.4) k0 k0 392 Homoclinic Chaos in Infinite-Dimensional Systems

C h Proof. a It follows from (5.2) that p t - = ()m ()()- + e x + m x + m Df z x x R ,(5.5) k0 k0 k0 k0 k0 k0 r where 6 R = gm ()x - g m ()x - ()Dfm ()z ()x - x . k0 k0 k0 k0 k0 k0 Since the set L is hyperbolic with respect to f , there exists a family of projectors {}Px(): x Î L for which equations (3.2)–(3.4) are valid. Therefore, m ()1- Pf()()z ()x +m - x +m = k0 k0 k0 = ()Df m z ()1- Pz()()x - x + ()1- Pf()m()z R . k0 k0 k0 k0 k0 k0 It means that ()1- Pz()()x - x = k0 k0 k0 m -1 m = []()Df z []1- Pf()()z []x +m - x +m - R . k0 k0 k0 k0 k0 Consequently, equation (3.4) implies that m ()1- Pz()()x - x £ Kq ()x +m - x +m + R . k0 k0 k0 k0 k0 k0 Let us estimate the value R . It can be rewritten in the form k0 1 R = []x()xDgm ()x + ()1- x x - Dfm ()z d × ()x - x . k0 ò k0 k0 k0 k0 k0 0 It follows from (5.3) that x x +e()1- x x - z £ . Hence, using (4.20) k0 k0 k0 and (4.21), for e > 0 small enough we obtain that

~~~~R £ ()rm()h + wm()e x - x ,(5.6) k0 k0 k0~~~~

where rm()h ®h0 as ®w0 and m()e ®e0 as ® 0 . Therefore, m ()1- Pz()()x - x £ Kq ()x +m - x +m + x - x ,(5.7) k0 k0 k0 k0 k0 k0 k0 r ()h + w()e £ - m provided m m 1 . Further, we substitute the value k0 for k0 in (5.5) to obtain that m x - x = Df ()z - m ()x - m - x - m + R - m . k0 k0 k0 k0 k0 k0 Therefore, using (3.2) we find that m Pf()()z -m ()x - x = k0 k0 k0 m m = Df ()z -m Pz()-m ()x -m - x -m + Pf()()z -m R -m . k0 k0 k0 k0 k0 k0 Birkhoff-Smale Theorem 393

-m Hence, equations (3.3) and (5.6) with k0 instead of k0 give us that m Pf()()z -m ()x - x £ k0 k0 k0 m £ Kq{}+ ()rm()h + wm()e x -m - x -m . (5.8) k0 k0 Since m - 1ìü - m ()= j()- j()() zk f zk -m å íýf zk - j f fzk - j -1 , 0 0 îþ0 0 j = 0 it follows from (4.15) and (5.1) that m -1 m j m z - f ()z - ££ddmM × ()1+ M k0 k0 m å j = 0 for d small enough. Therefore, m m Pz()-wdmPf()()z -m £ ()wdm× ()1 + M º (), , k0 k0 where wx()®x0 as ® 0 (cf. (4.22)). Consequently, estimate (5.8) implies that Pz()()x - x £ k0 k0 k0 m -1 £ Kq{}+++rm()h wm()e K × wd(), m x -m - x -m . (5.9) k0 k0 It is evident that estimates (5.7) and (5.9) enable us to choose the parameters mhe,,, and d such that x - x £ 1 max {}x - x : kJÎ ()k , m . k0 k0 2 k k 1 0

Using this inequality with kk instead of 0 we obtain that x - x £ 1 max {}x - x : nJÎ ()k , m k k 2 n n 2 0 Î (), m for all kJ1 k0 . Therefore, x - x £ 1 max {}x - x : nJÎ ()k , m . k0 k0 4 n n 2 0 If we continue to argue like that, then we find that -n x - x £ 2 max {}x - x : nJÎ n()k , m . k0 k0 n n 0 - £ - + - Since xn xn xn zn xn zn , this and estimate (5.3) imply (5.4). Lemma 5.1 is proved.

Proof of Theorem 5.1. ,,,¼ Let p1 p2 pm -1 be distinct integers. Let us choose and fix the parameters ehd,, >m0 and the integer > 0 such that (i) Theorem 4.2 and Lemma 5.1 L = {}È {}Î can be applied to the hyperbolic set x0 yn: n Z and (ii) 394 Homoclinic Chaos in Infinite-Dimensional Systems

C h ìü e < 1 - , - , , ¼, - , ¹ a min íýyp yp yp x0 : ij=1 m 1 i j .(5.10) p 2 îþi j i t e Assume that N is such that r d y - x < fornNp= ++1 andnp= - N (5.11) 6 n 0 2 i i = ,,¼ , - for all i 12 m 1 . Let us consider the segments Ci of the orbits of the mapping f of the form

C = ()y - ,,¼ y ,,¼ y + ,,i = 12,,¼ ,m -1 i pi N pi pi N = (),,,¼ Cm x0 x0 x0 . + = ()S¼ ¼ Î The length of every such segment is 2 N 1 . Let a a-1 a0 a1 m . Let us g consider a sequence of elements a made up of the segments Ci by the formula g = ()¼ C C C ¼ .(5.12) a a-1 a0 a1 g ÎgL d It is clear that a and by virtue of (5.11) a is a -pseudoorbit of the mapping { º () fw. Therefore, due to Theorem 4.2 there exists a unique trajectory n wn a : n Î}Z of the mapping g such that -e() £ wnN(),, i j zij a ,(5.13) (),, = ++()- + () where nN i j Ni1 2 N 1 j and zij a is the j -th element of the segment C , i Î ,j =jS12,,¼ ,2 N +1 . Let us define the mapping from ai Z m into X by the formula j()= a w0 ,(5.14) {} {} where w0 is the zeroth element of the trajectory wn . Since the trajectory wn possessing the property (5.13) is uniquely defined, equation (5.14) defines a map- S ping from m into X . If we substitute i +1 for i in (5.13) and use the equations

==2 N +1æö wnN(),, i+1 j wnN(),, i j ++2 N 1 g èøwnNij(),, , we obtain that 2 N +1 ()-e() £ g wnNij(),, zi +1, j a Î = ,,¼ , + ()= () for all i Z and j 12 2 N 1 . Therefore, the equality zi +1, j a zij Sa S with S being the Bernoulli shift in m leads us to the equation 2 N +1 ()-e()£ g wnNij(),, zi +1, j Sa . Consequently, the uniqueness property of the e -trajectory in Theorem 4.2 gives us the equation ()= 2 N +1 ()() Î wn Sa g wn a ,.n Z Birkhoff-Smale Theorem 395

This implies that j()= 2 N +1j() Î S Sa g a ,,a m (5.15) i.e. property (c) is valid for l = 2 N +1 . It follows from (5.15) that g2 N +1 ()jj()S-1a = ()a . = jS() 2N +1 Therefore, the set Y m is strictly invariant with respect to g . Thus, assertion (a) is proved. Let us prove the continuity of the mapping j . Assume that the sequence of ele- ()s =S()¼,,,,()s ()s ()s ¼ = ()Î¼, ,,,¼ ments a a-1 a0 a1 of m tends to a a-1 a0 a1 Î S ® ¥ Î m as s + . This means (see Exercise 1.4) that for any M N there exists = () s0 s0 M such that ()s = £ ³ ai ai foriM ,ss0 . (5.16) () g()s =g{}s =d{} L Assume that zk and zk are -pseudoorbits in constructed according to (5.12) for the symbols a()s and a , respectively. Equation (5.16) implies ()s = £ ()+ {} {}e()s that zk zk for kM2 N 1 . Let wn and wn be -trajectories corresponding to gg and ()s , respectively. Lemma 5.1 gives us that ()s - £ 1 -n e w0 w0 2 ,(5.17) ()nm+ ³nÎ ³ n, provided M 2N 1 , i.e. for any N equations (5.17) is valid for ss0 M . This means that j ()j()s - ()º ()s - ® a a w0 w0 0 ®j¥ = jS() as s + . Thus, the mapping is continuous and Y m is a compact strictly invariant set with respect to g2 N +1 . j ,S¢ Î Let us now prove nontriviality property (b) of the mapping . Let aa m ¹ ¢ ³ {} {}¢ e be such that ai ai for some i 0 . Let wn and wn be -trajectories corresponding to the symbols aa and ¢ , respectively. Then - ¢ ³ ()- ()¢ - wnNiN(),, +1 wnNiN(),, +1 ziN, +1 a ziN, +1 a - - ()- ¢ - ()¢ wnNiN(),, +1 ziN, +1 a wnNiN(),, +1 ziN, +1 a . () Therefore, it follows both from (5.13) and the definition of the elements zij a that w()+ - w ()¢ + ³ y - y ¢ - 2 e , 2N 1 i 2N 1 i qi qi where q = p andq ¢ = p ¢ . We apply (5.10) to obtain that i ai i ai - ¢ > w()2N +1 i w ()2N +1 i 0 .(5.18) Therefore, if i ³ 0 , then ()2N +1 i()¹ ()2N +1 i ()¢ g w0 g w0 . j()º ¹ ¢ º j()¢ Hence, a w0 w0 a . This completes the proof of assertion (b). 396 Homoclinic Chaos in Infinite-Dimensional Systems

C h If the trajectories of the mapping g cannot be “glued” (see the hypotheses of a Î ¹ ¢ Theorem 5.1), then for some i Z equation (5.18) gives us that w0 w0 , i.e. p j()¹ j()¢ ¹j¢ S t a a if aa . Thus, the mapping is injective in this case. Since m e is a compact metric space, then the injectivity and continuity of jj imply that is r a homeomorphism from S onto jS() . Theorem 5.1 is proved. 6 m m = jS() It should be noted that equations (5.13) and (5.14) imply that the set Y m lies in the eLh -vicinity of the hyperbolic set . Therewith, the values and l involved in the statement of the theorem depend on e and one can state that for any vicinity U of the set Lh there exist and l such that the conclusions of Theo- jS()Ì U = jS() rem 5.1 are valid and m . It is also clear that the set Y m is not uniquely determined.

Exercise 5.1 Assume that gf=j in Theorem 5.1. Prove that the mapping jS()É {}È {}³ can be constructed such that m x0 ynl: n 0 , where {}Î yn: n Z is a homoclinic orbit of the mapping f . Exercise 5.2 Prove the Birkhoff theorem: if the hypotheses of Theorem 5.1 hold, then for any e >h0 small enough there exist > 0 and l Î N such that for every mapping g Î Wh ()f there exist periodic trajectories of the mapping g l of any minimal period in the e -vicinity of the set L .

~~~~Exercise 5.3 Use Theorem 1.1 to describe all the possible types of behaviour of the trajectories of the mapping g on a set l = k ()jS() WgÈ m . k = 1~~~~

In conclusion, it should be noted that different infinite-dimensional versions of Ano- sov’s lemma and the Birkhoff-Smale theorem have been considered by many authors (see, e.g., [6], [10], [11], [12], and the references therein).

~~~~§ 6 Possibility of Chaos in the Problem of Nonlinear Oscillations of a Plate~~~~

In this section the Birkhoff-Smale theorem is applied to prove the existence of chaotic regimes in the problem of nonlinear plate oscillations subjected to a periodic load. The results presented here are close to the assertions proved in [13]. However, the methods used differ from those in [13]. Possibility of Chaos in the Problem of Nonlinear Oscillations of a Plate 397

Let us remind the statement of the problem. We consider its abstract version as in Chapter 4. Let HA be a separable Hilbert space and let be a positive operator {} with discrete spectrum in He , i.e. there exists an orthonormalized basis k in H such that = l <¼l £ l £l= ¥ Aek k ek ,,.0 1 2 lim n n ® ¥ The following problem is considered: ì \\ ++g \ 2 +() 12 2 - G + = w ï u u A u A u Au Lu h cos t , (6.1) í \ ï u = u Î F = DA(), u = u Î H . (6.2) î t = 0 0 1 t = 0 1 Here gG,, , and w are positive parameters, hHL is an element of the space , is a linear operator in H subordinate to A , i.e. Lu£ K Au ,(6.3) where K is a constant. The problem of the form (6.1) and (6.2) was studied in Chap- ter 4 in details (nonlinearity of a more general type was considered there). The results of Section 4.3 imply that problem (6.1) and (6.2) is uniquely solvable in the class of functions W = (), ()Ç 1(), + C R+ DA C R+ H .(6.4) Moreover, one can prove (cf. Exercise 4.3.9) that Cauchy problem (6.1) and (6.2) is uniquely solvable on the whole time axis, i.e. in the class W = C()R, DA()Ç C1()R, H . This fact as well as the continuous dependence of solutions on the initial conditions (see (4.3.20)) enables us to state that the monodromy operator G acting in H = = DA()´ H according to the formula

(); = æöæö2p , \ æö2p Gu0 u1 èøu èøw u èøw (6.5) is a homeomorphism of the space H (see Exercise 4.3.11). Here ut() is a solution to problem (6.1) and (6.2) The aim of this section is to prove the fact that under some conditions on L and h chaotic dynamics is observed in the discrete dynamical system ()H , G for some set of parameters gG,, , and w .

~~~~Lemma 6.1. The mapping G defined by equality (6.5) is a diffeomorphism of the space H = DA()´ H .~~~~

Proof. () We use the method applied to prove Lemma 4.7.3. Let u1 t be a solution = (), Î H to problem (6.1) and (6.2) with the initial conditions yu0 u1 and let 398 Homoclinic Chaos in Infinite-Dimensional Systems

C () + º ()+ , + ÎH h u2 t be a solution to it with the initial condions yz u0 z0 u1 z1 . a Let us consider a linearization of problem (6.1) and (6.2) along the solution u ()t : p 1 t ì \\ ()+++ g\ () 2 () 12 ()2 - G + e ï w t w t A w A u1 t Aw r ï +()12 (), 12 () ()+ í 2 A u1 t A wt Au1 t Lw = 0 , (6.6) 6 ï \ ï w = z , w = z . (6.7) î t = 0 0 t = 0 1 As in the proof of Theorem 4.2.1, it is easy to find that problem (6.6) and (6.7) is ()= ()- ()- () uniquely solvable in the class of functions (6.4). Let vt u2 t u1 t wt . It is evident that vt() is a weak solution to problem

ì \\ ++g \() 2 ()= ()º ()(),,() () ï v v t A vt Ft Fu1 t u2 t wt , (6.8) í \ ï v ==0, v 0 , (6.9) î t = 0 t = 0 where (),, = -æö12 2 - G + æö 12 2 - G Fu1 u2 w èøA u2 Au2 èøA u1 Au1 +

++æö 12 2 - G (), - ()- - èøA u1 Aw 2 Au1 w Au1 Lu2 u1 w . A simple calculation shows that (),, = (),, + (),, º + Fu1 u2 w F1 u1 u2 w F2 u1 u2 w F1 F2 , where = -æö 12 2 - G - - (), ()+ F1 èøA u2 Av Lv 2 Au1 v Au1 w ,

~~~~= - 12 ()- 2 ()+ - (), F2 A u1 u2 Au1 w 2 Au1 w Aw .~~~~

We assume that y H £ R and z H £ 1 . In this case (see Section 4.3) the estimates () £ Auj t CRT, , ()()- () £ Au1 t u2 t CRT, z H , () £ Aw t CRT, z H ,(6.10) [], are valid on any segment 0 T . Here CRT, is a constant. Therefore, £ £ 2 Î [], F1 C1 Av , F2 C2 z H ,t 0 T ,

~~~~where C1 and C2 are constants depending on R and T. Hence, ()2 £ 2 + 4 Î [], Ft C1 Av C2 z H ,.t 0 T Possibility of Chaos in the Problem of Nonlinear Oscillations of a Plate 399~~~~

Therefore, the energy equation t t 1 æö\ 2 2 \ 2 \ v()t + Av() t + g v()t dt = ()tF()t , v()t d (6.11) 2 èøò ò 0 0 for problem (6.8) and (6.9) leads us to the estimate t \()2 + ()2 £ æö()t 2 + 4 t Î [], v t Av t òèøC1 Av C2 z d ,.t 0 T 0 Using Gronwall’s lemma we find that \ v()t 2 + Av() t 2 £ Cz4 ,,t Î []0, T where the constant CTR depends on and . This estimate implies that the mapping

~~~~¢ º (); ® æöæö2p ; \ æö2p G : zz0 z1 èøw èøw wèøw (6.12)~~~~

is a Frechét derivative of the mapping Gwt defined by equality (6.5). Here () is a solution to problem (6.6) and (6.7). It follows from (6.10) and (6.6) that G¢ is a continuous linear mapping of H into itself. Using (6.10) it is also easy to see ¢ = ¢[], = (); that G G u0 u1 continuously depends on yu0 u1 with respect to the operator norm. Lemma 6.1 is proved.

~~~~Further we will also need the following assertion.~~~~

~~~~Lemma 6.2.~~~~

Let Gj be the monodromy operator of problem (6.1) and (6.2) with = = = , = LLj and hhj , j 12 . Assume that for LLj equation (6.3) is va- Î = , lid and hj H , j 12 . Moreover, assume that -1 £ r £ r = , Lj A ,,.hj j 12 Then the estimates æö sup G ()y - G ()y £ CL()- L A-1 + h - h (6.13) Î 1 2 èø1 2 1 2 yBR and æö sup G ¢ ()y - G¢()y £ CL()- L A-1 + h -h (6.14) Î 1 2 èø1 2 1 2 yBR H = ()´ > are valid. Here BR is a ball of the radius R in DA H , R 0 is an arbitrary number while the constant CR depends on and r but does not depend on the parameters wg,, ³G0 , and provided they vary in bounded sets. 400 Homoclinic Chaos in Infinite-Dimensional Systems

C h Proof. a Let u ()t be a solution to problem (6.1) and (6.2) with LL= and hh= , p j j j t j = 12, . It is evident (see Section 4.3) that e r () £ º (),,r Î [], Auj t CCR T ,.(6.15)t 0 T 6 ()= ()- () Therefore, it is easy to find that the difference ut u1 t u2 t satisfies the equation ì \\ ++g 2 = (),, ï u uAu Ft u1 u2 , í \ ï u ==0, u 0 , î t = 0 t = 0 (),, where the function Ft u1 u2 can be estimated as follows: (),, £ + æö()- -1 + - Ft u1 u2 C1 Au C2 èøL1 L2 A h1 h2 . As in the proof of Lemma 6.1, we now use energy equality (6.11) and Gronwall’s lemma to obtain the estimate \ ()2 + ()2 £ æö()- -1 2 + - 2 u t Au t CLèø1 L2 A h1 h2 .(6.16) This implies inequality (6.13). Estimate (6.14) can be obtained in a similar way. In its proof equations (6.12), (6.15), and (6.16) are used. We suggest the reader to carry out the corresponding reasonings himself/herself. Lemma 6.2 is proved.

Let us now prove that there exist an operator Lh and a vector such that the corresponding mapping G possesses a hyperbolic homoclinic trajectory. To do that, we use the following well-known result (see, e.g., [1], [13], as well as Section 7) related to the Duffing equation.

Theorem 6.1. Let g : R2 ® R2 be a monodromy operator corresponding to the Duffing equation \\ \ x -abx +ex3 = ()f × cosw t - dx , (6.17) i.e. the mapping of the plane R2 into itself acting according to the formula

(), = æöæö2p ; \ æö2p gx0 x1 èøx èøw x èøw , (6.18) () ()= \ ()= where xt is a solution to equation (6.17) such that x 0 x0 and x 0 = x1 . All the parameters contained in (6.17) are assumed to be positive. Let us also assume that b32 pw > º d ××2 æö ffcr coshèø. (6.19) 3w 2 a 2 b Possibility of Chaos in the Problem of Nonlinear Oscillations of a Plate 401

e >eÎ (), e Then there exists 0 0 such that for every 0 0 the mapping g pos- {}Î ¹ sesses a fixed point zy and a homoclinic trajectory n: n Z to it, yn z , {}È {}Î therewith the set z yn: n Z is hyperbolic.

Let Pk be the orthoprojector onto the one-dimensional subspace generated by the = ()- eigenvector ek in HL . We consider problem (6.1) and (6.2) with k L 1 Pk in- = × stead of Lhh and k ek , where hk is a positive number. Then it is evident that = every solution to problem (6.1) and (6.2) with the initial conditions u0 c0 ek and = u1 c1 ek has the form () = () ut xt ek , where xt() is a solution to the Duffing equation \\ ++g \ -ll ()Gl- 2 3 = w x x k k x k x hk cos t (6.20) ()= \ ()= with the initial conditions x 0 c0 and x 0 c1 . In particular, this means that L = {}(); , (); H the two-dimensional subspace k Lin ek 0 0 ek of the space is strictly invariant with respect to the corresponding monodromy operator Gk while the re- L striction of Gk to k coincides with the monodromy operator corresponding to the Duffing equation (6.20). Therefore, if hk is small enough and the conditions h 2 l ()Gl- æöG 12 pw 0 < k k - 1 cosh k g w èøl l ()Gl- 3 2 k 2 k k hold, then the mapping Gk possesses a hyperbolic invariant set L = {}È {}× Î k zek yn ek : n Z (); consisting of the fixed point z0 ek z1 ek and its homoclinic trajectory {}Î = ()0 ; 1 Î 2 yn ek : n Z ,whereyn yn yn R . w > <

Theorem 6.2. w > < P ´ exist 0 and an open set in the metric space R+ H such that if l-1 < m ()g, Î P Lek k ,,,h , then some degree Gl of the monodromy operator G of problem (6.1) and (6.2) possesses a compact strictly invariant set Y ()Gl Y = Y in the space H in which the mapping Gl is topologically conjugate to the Bernoulli shift of 402 Homoclinic Chaos in Infinite-Dimensional Systems

C j S ® h sequences of m symbols, i.e. there exists a homeomorphism : m Y a such that p t Gl ()jj ()a = ()Sa ,,.a Î S . e m r 6 Exercise 6.1 Prove that if the hypotheses of Theorem 6.2 hold, then equation (6.1) possesses an infinite number of periodic solutions with pe- riods multiple to w .

~~~~Exercise 6.2 Apply Theorem 6.2 to the Berger approximation of the problem of nonlinear plate oscillations:~~~~

ì ¶2u ¶u æö ï ++g D2u - ç÷Ñux(), t 2 dx - G Du + ï ¶t2 ¶t èøò ï W ï ¶ í r u () w = ()W, Î Ì 2 > + ¶ = hx cos t , xx1 x2 R , t 0 , ï x1 ï ï ¶u u ==Du 0, u =u ()x , =u ()x . ï ¶W ¶W t = 0 0 ¶t 1 î t = 0

~~~~§ 7 On the Existence of Transversal Homoclinic Trajectories~~~~

Undoubtedly, Theorem 6.1 on the existence of a transversal (hyperbolic) homoclinic trajectory of the monodromy operator for the periodic perturbation of the Duffing equation is the main fact which makes it possible to apply the Birkhoff-Smale theorem and to prove the possibility of chaotic dynamics in the problem of plate oscillations. In this connection, the question as to what kind of generic condition guarantees the existence of a transversal homoclinic orbit of monodromy operators generated by ordinary differential equations gains importance. Extensive literature is devoted to this question (see, e.g., [1], [2] and the references therein). There are several approaches to this problem. All of them enable us to construct systems with transversal homoclinic trajectories as small perturbations of “simple” systems with homoclinic (not transversal!) orbits. In some cases the corresponding conditions on perturbations can be formulated in terms of the Melnikov function. This section is devoted to the exposition and discussion of the results obtained by K. Palmer [14]. These results help us to describe some classes of systems of ordinary differential equations which generate dynamical systems with transversal ho- On the Existence of Transversal Homoclinic Trajectories 403 moclinic orbits. Such differential equations are obtained as periodic perturbations of autonomous equations with homoclinic trajectories. In the space Rn let us consider a system of equations \ x()t = gxt()() ,,xt()Î Rn (7.1) where g : Rn ® Rn is a twice continuously differentiable mapping. Assume that the Cauchy problem for equation (7.1) is uniquely solvable for any initial condition ()= ()() x 0 x0 . Let us also assume that there exist a fixed point z0 gz0 =0 and ()¹ a trajectory zt z0 homoclinic to z0 , i.e. a solution to equation (7.1) such that zt()® z as t ® ±¥ . Exercises 7.1 and 7.2 given below give us the examples of the 0 \\ cases when these conditions hold. We remind that every second order equation x + + ()= = Vx 0 can be rewritten as a system of the form (7.1) if we take x1 x and = \ x2 x .

Exercise 7.1 Consider the Duffing equation \\ x -ab x + x3 =ab0 ,., > 0 \ Prove that the curve zt()= () h()t , h()t ÎR2 is an orbit of the corresponding system (7.1) homoclinic to 0 . Here h()t = 2 ba¤bsech t . Exercise 7.2 Assume that for a function Ux()Î C3()R there exist a number Eab and a pair of points < such that Ua()==Ub() E ;,;Ux()< E xabÎ (), U¢()a = 0 ;;.U¢¢()a < 0 U¢()b > 0 \\ Then system (7.1) corresponding to x + U¢()x = 0 possesses an orbit homoclinic to ()a, 0 that passes through the point ()b, 0 .

Unfortunately, as the cycle of Exercises 7.3–7.5 shows, the homoclinic orbit of autonomous equation (7.1) cannot be used directly to construct a discrete dynamical system with a transversal homoclinic trajectory.

Exercise 7.3 For every t > 0 define the mapping f : Rn ® Rn by the for- ()= ()t ; (); mula fx0 x x0 , where xt x0 is a solution to equation (7.1) with the initial condition x0 . Show that f is a diffeomorphism n in R with a fixed point z0 and a family of homoclinic orbits {}s Î = ()+ t Î [ , t) yn : n Z , where yn zs n , s 0 . Exercise 7.4 Prove that the derivative f ¢ of the mapping f constructed in ¢() = ()t, Exercise 7.3 can be evaluated using the formula f x0 wy w , where yt(), w is a solution to problem \ ()= ¢()() () ()= y t g xt yt ,.y 0 w0 ()= (); Here xt xt x0 is a solution to equation (7.1) with the initial condition x0 . 404 Homoclinic Chaos in Infinite-Dimensional Systems

C h Exercise 7.5 Let f be the mapping constructed in Exercise 7.3 and let a {}zt(): t Î Z be a homoclinic orbit of equation (7.1). Show that p \ t {}()t Î wn = z n : n Z is a bounded solution to the difference equa- e = ¢() = ()t r tion wn +1 f yn wn , where yn zn (Hint: the function ()= \() \ = ¢()() 6 wt z t satisfies the equation w g zt w ). Thus, due to Theorems 2.1 and 3.1 the result of Exercise 7.5 implies that the set L = {}È {}()t Î z0 zn : n Z cannot be hyperbolic with respect to the mapping f defined by the formula ()= ()t ; (); fx0 x x0 , where xt x0 is a solution to equation (7.1) with the initial condition x0 . Nevertheless we can indicate some quite simple conditions on the class of perturbations {}ht(),, x m periodic with respect to t under which the monodromy operator of the problem \ x()t = gxt()m() + ht(),, xt() m ,,(7.2)xt()Î Rn possesses a transversal (hyperbolic) homoclinic trajectory for m small enough.

Further we will use the notion of exponential dichotomy for ordinary differential equations (see [15], [16] as well as [5] and the references therein) Let At() be a continuous and bounded nn´ matrix function on the real axis. We consider the problem \ x()t = At()xt(),,t Î x = x , (7.3) R ts= 0 in the space X = Rn . It is easy to see that it is solvable for every initial condition. Therefore, we can define the evolutionary operator $()ts, , ts, Î R , by the formula $(), = ()º (), ; , Î tsx0 xt xt s x0 ,,ts R where xt() is a solution to problem (7.3).

~~~~Exercise 7.6 Prove that $()ts, =$ $()$tt, t (), s , ()tt, = I for all ts,,t Î R and the following matrix equations hold: d $()ts, = At()$()ts, ,.(7.4)d $()ts, = - $()ts, As() dt ds~~~~

~~~~Exercise 7.7 Prove the inequality~~~~

~~~~ìüt ïï $()ts, £ exp íýò A()t d t ,.ts³ ïï îþs On the Existence of Transversal Homoclinic Trajectories 405~~~~

Let I be some interval of the real axis. We say that equation (7.3) admits an exponential dichotomy over the interval I if there exist constants K, a > 0 and a family of projectors {}Pt(): t ÎI continuously depending on t and such that Pt()$()$ts, = ()ts, Ps(),;ts³ (7.5) $()ts, Ps() £ Ke-a ()ts- ,,ts³ (7.6) -a ()- $()ts, ()IPs- () £ Ke st,,ts£ (7.7) for ts, ÎI .

~~~~Exercise 7.8 Let At()º A be a constant matrix. Prove that equation (7.3) admits an exponential dichotomy over R if and only if the eigenvalues on A do not lie on the imaginary axis.~~~~

~~~~The assertion contained in Exercise 7.8 as well as the following theorem on the roughness enables us to construct examples of equations possessing an exponential dichotomy.~~~~

Theorem 7.1. Assume that problem (7.3) possesses an exponential dichotomy over an interval I . Then there exists e > 0 such that equation \ x()t = ()At()+ Bt() xt() (7.8) possesses an exponential dichotomy over I , provided Bt()£ e for t Î I . Moreover, the dimensions of the corresponding projectors for (7.3) and (7.8) are the same.

~~~~The proof of this theorem can be found in [15] or [16], for example.~~~~

~~~~The exercises given below contain some simple facts on systems possessing an exponential dichotomy. We will use them in our further considerations.~~~~

~~~~Exercise 7.9 Prove that equations (7.5)–(7.7) imply the estimates a ()- $()xts, ³ K -1 e ts()x1- Ps() ,,ts³ a ()- $()xts, ³ K -1 e st Ps()x,,ts£ for any x ÎX º Rn .~~~~

Exercise 7.10 Assume that equation (7.3) admits an exponential dichotomy = [ , ¥) () = over R+ 0+ . Prove that P 0 XV+ , where ìü º x Î º n $()x, < ¥ V+ íýX R : sup t 0 (7.9) îþt > 0 (Hint: $()xt, 0 ³ K-1 ea t ()x1- P()0 ,t ³ 0 ). 406 Homoclinic Chaos in Infinite-Dimensional Systems

C h Exercise 7.11 If equation (7.3) possesses an exponential dichotomy over a = (-¥, ] ()- () = p R– 0 , then IP0 XV– , where t ìü e = x Î º n $()x, < ¥ r V– íýX R : sup t 0 (7.10) îþt < 0 6 (Hint: $()xt, 0 ³ K -1 ea t P()x0 ,t £ 0 ).

~~~~Exercise 7.12 Assume that equation (7.3) possesses an exponential dicho-~~~~

~~tomy over R+ (over R– , respectively). Show that any solution to problem (7.3) bounded on R+ (on R– , respectively) decreases at exponential velocity as t ® + ¥ (as t ® -¥ , respectively).~~

Exercise 7.13 Assume that equation (7.3) possesses an exponential dichotomy over the half-interval [a, + ¥) , where a is a real number. Prove that equation (7.3) possesses an exponential dichotomy over any semiaxis of the form [b, + ¥) . (Hint: Pt()= $()ta, Pa()$()at, ).

~~~~Exercise 7.14 Prove the analogue of the assertion of Exercise 7.13 for the semiaxis (-¥, a] .~~~~

Exercise 7.15 Prove that for problem (7.3) to possess an exponential dichotomy over R it is necessary and sufficient that equation (7.3) possesses an exponential dichotomy both over R+ and R– and has no nontrivial solutions bounded on the whole axis R .

Exercise 7.16 Prove that the spaces V+ and V– (see (7.9) and (7.10)) possess the properties n V+ Ç V– = {}0 ,,V+ + V– = X º R provided problem (7.3) possesses an exponential dichotomy over R .

Exercise 7.17 Consider the following equation adjoint to (7.3): \ y()t = -A*()t yt(), (7.11) where A*()t is the transposed matrix. Prove that the evolutionary operator Y()ts, of problem (7.11) has the form Y()$ts, = []()st, * .

Exercise 7.18 Assume that problem (7.3) possesses an exponential dichotomy over an interval I . Then equation (7.11) possesses exponential dichotomy over I with the same constants K, a > 0 and projectors Qt()= IPt- ()* .

Exercise 7.19 Assume that problem (7.3) possesses an exponential dichoto- + = my both over R+ and R– . Let dimV+ dimV– n , where V± are defined by equalities (7.9) and (7.10). Show that the dimensions of the spaces of solutions to problems (7.3) and (7.11) bounded on the whole axis are finite and coincide. On the Existence of Transversal Homoclinic Trajectories 407

Exercise 7.20 Assume that problem (7.3) possesses an exponential dichotomy over R . Then for any s ÎtR and > 0 the difference equation = $()+ t , xn s n s xn -1 possesses an exponential dichotomy over Z (for the definition see Section 2).

Let us now return to problem (7.1). Assume that z0 is a hyperbolic fixed point for ¢() (7.1), i.e. the matrix g z0 does not have any eigenvalues on the imaginary axis. Let () zt be a trajectory homoclinic to z0 . Using Theorem 7.1 on the roughness and the results of Exercises 7.13 and 7.14 we can prove that the equation \ y = g ¢()zt() y (7.12) possesses an exponential dichotomy over both semiaxes R+ and R– . Moreover, the dimensions of the corresponding projectors are the same and coincide with the di- ¢() mension of the spectral subspace of the matrix g z0 corresponding to the spectrum in the left semiplane. Therewith it is easy to prove that dimV+ + dimV– = n , where V± have form (7.9) and (7.10). The result of Exercise 7.15 implies that equa- \ tion (7.12) cannot possess an exponential dichotomy over R (yt()= z()t is a solution to (7.12) bounded on R ) while Exercise 7.19 gives that the number of linearly independent bounded (on R ) solutions to (7.12) and to the adjoint equation \ y = -[]g ¢()zt() * y (7.13) is the same. These facts enable us to formulate Palmer’s theorem (see [14]) as follows.

Theorem 7.2. Assume that gx() is a twice continuously differentiable function from Rn into Rn and equation \ x = gx() {}() Î possesses a fixed hyperbolic point z0 and a trajectory zt: t R homo- ()= \() clinic to z0 . We also assume that yt z t is a unique (up to a scalar factor) solution to equation \ y = g ¢()zt() y (7.14) bounded on R . Let ht(),, x m be a continuously differentiable vector func- Î -D()< tion Ttt-periodic with respect to and defined for R , xzt 0 , ms< m Î 0 , R . If ¥ ¥ ()y()t , ht(),, zt() 0 dt =y0 ,,,()()t , h ()tzt,,() 0 dt ¹ 0 , (7.15) ò Rn ò t Rn -¥ -¥ where y()t is a bounded (unique up to a constant factor) solution to the equation adjoint to (7.14), then there exist Ds and and suchsuch that that for for 0 <

C h possesses the following properties: a (a) there exists a unique T -periodic solution x ()t, m such that p 0 t z -Dx ()t, m < ,,,t Î , e 0 0 R r and 6 - x (), m ® m ® supz0 0 t 0 ,,;0 ; t Î R (b) there exists a solution x ()t, m bounded on R and such that x ()t, m -Dzt() < ,,,t Î R ,

supx ()t, m - zt() = 0()m , t Î R and x()x, m - (), m = limt 0 t 0 ; t ® ±¥ (c) the linearized equation \ ìü y = íýg ¢h()m()t, m + h¢ ()t,,h()mt, m y , (7.17) îþx h(), m x(), m x(), m where t is equal to either t or 0 t , possesses an exponential dichotomy over R . This theorem immediately implies (see Exercise 7.20 and Theorem 3.1) that under conditions (7.15) the monodromy operator for problem (7.16) has a hyperbolic fixed point in a vicinity of the orbit {}zt(): t Î R and a transversal trajectory homoclinic to it.

We will not prove Theorem 7.2 here. Its proof can be found in paper [14]. We only outline the scheme of reasoning which enables us to construct a homoclinic trajectory x ()t, w . Here we pay the main attention to the role of conditions (7.15). If we change the variable xzt= ()+ z in equation (7.16), then we obtain the equation \ z = gzt()()+ z -mgzt()() + ht(),, zt()+m z . 1(), n ´ We use this equation to construct a mapping from Cb RR R into 0 (), n Cb RR acting according to the formula \ ()zm, ®z ()zm, = - {}gzt()()+ z -mgzt()() + ht(),, zt()+m z .(7.18) k(), We remind that Cb R X is the space of k times continuously differentiable bounded functions from R into Xtk with bounded derivatives with respect to up to the - th order, inclusive. Thus, the existence of bounded solutions to problem (7.16) is equivalent to the solvability of the equation ()zm, = 0 . It is clear that ()00, = 0 . Therefore, in order to construct solutions to equation ()zm, = 0 we should apply an appropriate version of the theorem on implicit functions. Its standard statement requires that the operator LD= z ()00, be invertible. However, it is easy to check that the operator L has the form On the Existence of Transversal Homoclinic Trajectories 409

\ ()Ly ()t = y ()t - g ¢()zt() yt() . \ Therefore, it possesses a nonzero kernel ()Lz= 0 . Hence, we should use the modified (nonstandard) theorem on implicit functions (see Theorem 4.1 in [14]). Roughly speaking, we should make one more change zm= w and consider the equation ()m w, m = 0 .(7.19) If this equation is solvable and the solution w depends on m smoothly, then w satisfies the equation

Dz ()m w, m []w + m wm + Dm ()m w, m = 0 ,(7.20) where wm is the derivative of w with respect to the parameter m . This equation can be obtained by differentiation of the identity ()m w ()mm , =m0 with respect to . Due to the smoothness properties of the mapping , it follows from (7.20) the solvability of the problem (), + (), = Dz 00w0 Dm 00 0 ,(7.21) which is equivalent to the differential equation \ = ¢()() + (),,() w0 g zt w0 ht zt 0 (7.22) in the class of bounded solutions. It is easy to prove that the first condition in (7.15) is necessary for the solvability of (7.22) (it is also sufficient, as it is shown in [14]). Further, the necessary condition of the dichotomicity of (7.17) for h()=t, m = x ()t, m on R is the condition of the absence of nonzero solutions to equation (7.17) bounded on R . However, this equation can be rewritten in the form

Dz ()m w, m y()m = 0 ,(7.23) where ww= ()m is determined with (7.19). If we assume that equation (7.23) has nonzero solutions, then we differentiate equation (7.23) with respect to m at zero, as above, to obtain that (), + (), + (), = Dzz 00w0 Dzm 00 y0 Dz 00y1 0 ,(7.24) = () = () ()m where w0 is a solution to equation (7.21), y0 y 0 ,y1 Dm y 0 , and y is a solution to equation (7.23). Equation (7.17) transforms into (7.14) when m = 0 . Therefore, the condition of uniqueness of bounded solutions to (7.14) gives us that = \() = y0 c0 z t , therewith we can assume that c0 1 . Hence, equation (7.24) transforms into an equation for y1 of the form \ - ¢()() = () y1 g zt y1 at ,(7.25) where

~~~~() ==-æö (), + (), \ () at èøDzz 00w0 Dzm 00 z t~~~~

~~~~= ()() ()+ (),,() \() Dxx gzt w0 t Dx ht zt 0 z t . 410 Homoclinic Chaos in Infinite-Dimensional Systems~~~~

C () h Here w0 t is a solution to (7.22). A simple calculation shows that equation (7.25) a can be rewritten in the form p t d ()()+ \ () - ¢()() ()()+ \ () = (),,() e y1 t w0 t g zt y1 t w0 t ht tzt 0 .(7.26) r dt 6 The second condition in (7.15) means that equation (7.26) cannot have solutions bounded on the whole axis. It follows that equation (7.23) has no nonzero solutions, i.e. equation (7.17) is dichotomous for h()xt, m = ()t, m . Thus, the first condition in (7.15) guarantees the existence of a homoclinic trajectory x()t, m while the second one guarantees the exponential dichotomicity of the linearization of the equation along this trajectory. x (), m As to the existence and properties of the periodic solution 0 t , this situation is much easier since the point z0 is hyperbolic. The standard theorem on implicit functions works here. It should be noted that condition (7.15) can be modified a little. If we consider a ()= ()- Î () “shifted” homoclinic trajectory zs t zt s for s R instead of zt in Theo- rem 7.2, then the first condition in (7.15) can be rewritten in the form ¥ D()s º æöy()ts- , ht(),, zt()- s 0 dt = 0 . ò èøRn -¥ If we change the variable tts® + , then we obtain that ¥ æö Dy()s = ()t , ht()+ s,,zt() 0 dt .(7.27) ò èøRn -¥ It is evident that ¥ æö D¢()s = y()ts- , h ()tzts,,()- 0 dt . ò èøt Rn -¥ Therefore, the second condition in (7.15) leads us to the requirement D¢()s ¹ 0 . D() D()=D¢()¹ Thus, if the function s has a simple root s0 (s0 0 ,s0 0 ), then the ()- () assertions of Theorem 7.2 hold if we substitute the value zt s0 for zt in (b). Performing the corresponding shift in the function x ()t, m , we obtain the assertions of the theorem in the original form. Thus, condition (7.15) is equivalent to the requirement D()=D¢()¹ Î s0 0 ,s0 0 for somes0 R , (7.28) where D()s has form (7.17).

In conclusion we apply Theorem 7.2 to prove Theorem 6.1. The unperturbed Duffing equation can be rewritten in the form \ = ì x1 x2 , í (7.29) î \ = b - a 3 x2 x1 x1 . On the Existence of Transversal Homoclinic Trajectories 411

\ The equation linearized along the homoclinic orbit zt()= () h()t , h()t (see Exer- cise 7.1) has the form \ = ì y1 y2 , í (7.30) î \ = b - ah()2 y2 y1 3 t y1 . Let us show that system (7.30) has no solutions which are bounded on the axis and \ \ not proportional to z()t . Indeed, if wt()= ()vt(), v()t is another bounded solution, \ \\ then due to the fact that h()t + h()t ® 0 as t ® ¥ , the Wronskian Wt()= \\ \ \ = vt()h()t - v()ht ()t possesses the properties

d Wt()= 0 andlimWt()= 0 . dt t ® ¥ \ This implies that Wt()º 0 and therefore vt() is proportional to h()t . Evidently, the equation adjoint to (7.30) has the form \ = - b + ah()2 ì y1 y2 3 t y2 , í (7.31) î \ = - y2 y1 . Since we have that \\ - b + ah()2 = y2 y2 3 t y2 0 , \\ \ a solution to (7.31) bounded on R has the form y()t = ()- h()t ; h()t . Let us now consider the corresponding function D()s . Since in this case htx()=,,m = (); w - d 0 f cos t x2 , we have ¥ \ \ Dh()s = ò ()t []f cosw()ts+ - dh()t dt , -¥ where b b -1 h() ==2 b 8 æöb t + - b t t a sech t a èøe e . Calculations (try to do them yourself) give us that w D()= w 2 × sin s - 4 db32 s 2 f a a . æöpw 3 coshèø 2 b Therefore, equation D()s = 0 has simple roots under condition (6.19). Thus, the assertion of Theorem 6.1 follows from Theorem 7.2. It should be noted that in this case the function D()s coincides with the famous Melnikov function arising in the geometric approach to the study of the transversality (see, e.g., [1], [2] and the references therein). Therewith conditions (7.28) transform into the standard requirements on the Melnikov function which guarantee the appearance of homoclinic chaos. 412 Homoclinic Chaos in Infinite-Dimensional Systems

C h Addition to the English translation: a p The monographs by Piljugin [1*] and by Palmer [2*] have appeared after publication t e of the Russian version of the book. Both monographs contain an extensive bibliogra- r phy and are closely related to the subject of Chapter 6. 6 References

1. GUCKENHEIMER J., HOLMES P. Nonlinear oscillations, dynamical systems and bifurcations of vector fields. –New York: Springer, 1983. 2. WIGGINS S. Global bifurcations and chaos: analytical methods.–New York: Springer, 1988. 3. NITECKI Z. Differentiable dynamics. –Cambrige: M.I.T. Press, 1971. 4. PALMER K. J. Exponential dichotomies, the shadowing lemma and transversal homoclinic points // Dynamics Reported. –1988. –Vol. 1. –P. 265–306. 5. HENRY D. Geometric theory of semilinear parabolic equations. Lect. Notes in Math. 840. –Berlin: Springer, 1981. 6. CHOW S.-N., LIN X.-B., PALMER K. J. A shadowing lemma with applications to semilinear parabolic equations // SIAM J. Math. Anal. –1989. –Vol. 20. –P. 547–557. 7. NICOLIS G., PRIGOGINE I. Exploring complexity. An introduction. –New York: W. H. Freeman and Company, 1989. 8. KATO T. Perturbation theory for linear operators. –Berlin; Heidelberg; New York.: Springer, 1966. 9. RUDIN W. Functional analysis. 2nd ed. –New York: McGraw Hill, 1991. 10. LERMAN L. M ., SHILNIKOV L. P. Homoclinic structures in infinite-dimentional systems // Siberian Math. J. –1988. –Vol. 29. –#3. –P. 408–417. 11. HALE J. K., LIN X.-B. Symbolic dynamics and nonlinear semiflows // Ann. Mat. Pura ed Appl. –1986. –Vol. 144. –P. 229–259. 12. LANI-WAYDA B. Hyperbolic sets, shadowing and persistance for noninvertible mappings in Banach spaces. –Essex: Longman, 1995. 13. HOLMES P., MARSDEN J. A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam // Arch. Rat. Mech Anal. –1981. –Vol. 76. –P. 135–165. 14. PALMER K. J. Exponential dichotomies and transversal homoclinic points // J. Diff. Eqs. –1984. –Vol. 55. –P. 225–256. 15. HARTMAN P. Ordinary differential equations. –New York; London; Sydney: John Wiley, 1964. 16. DALECKY Y. L., KREIN M. G. Stability of solutions of differential equations in Banach space. –Providence: Amer. Math. Soc., 1974.

1*. PILJUGIN S. JU. Shadowing in dynamical systems. Lect. Notes in Math. 1706. –Berlin; Heidelberg; New York: Springer, 1999. 2*. PALMER K. Shadowing in dynamical systems. Theory and Applications. –Dordrecht; Boston; London: Kluwer, 2000.

~~~~Index~~~~

@-pseudotrajectory 381 A-trajectory 382 A Attractor fractal exponential 61, 103 global 20 minimal 21 regular 39 weak 21, 32 B Belousov-Zhabotinsky equations 334 Berger equation 217 Bernoulli shift 14, 365 Burgers equation 96 C Cahn-Hilliard equation 97 Cantor set 52 Chaotic dynamics 369 Chirikov mapping 26 Completeness defect 296 Cycle 17 D Determining functionals 286, 292 boundary 358 mixed 332 Determining nodes 313, 332 Diameter of a set 53 Dichotomy exponential 370, 405 Dimension of a set fractal 52 Hausdorff 54 Dimension of dynamical system 11 Duffing equation 12, 26, 37, 400 Dynamical system = -dissipative 104 asymptotically compact 26, 249 asymptotically smooth 34 compact 27 continuous 11 discrete 11, 14, 365 dissipative 24 pointwise dissipative 34 416 Index

E Equation nonlinear diffusion, explicitly solvable 118 nonlinear heat 94, 117, 176, 187, 358 oscillations of a plate 217, 402 reaction-diffusion 96, 180, 328 retarded 13, 138, 171, 327 second order in time 189, 200, 209, 217, 350 semilinear parabolic 77, 85, 182, 317 wave 189, 201, 208, 351, 360 Evolutionary family 193, 321 Evolutionary semigroup 11 F Flutter 217 Frechét derivative 377 Fredholm operator 113 Function Bochner integrable 219 Bochner measurable 218 G Galerkin approximate solution 88, 190, 224, 233 Galerkin method non-linear 209 traditional 87, 190, 224, 233 Generic property 117 Gevrey regularity 347 Green function of difference equation 372 H Hausdorff metric 32, 48 Hodgkin-Huxley equations 339 Hopf model of appearance of turbulence 40, 130 I Ilyashenko attractor 22 Inertial form 152 Inertial manifold 151 approximate 182, 200, 276 asymptotically complete 161 exponentially asymptotically complete 161 local 171, 276 Invariant torus 138 K Kolmogorov width 302 L Landau-Hopf scenario 138 Lorentz equations 28 Lyapunov function 36, 108 Lyapunov-Perron method 153 Index 417

M Mapping continuously differentiable 377 Frechét differentiable 113, 377 proper 113 Melnikov function 411 Milnor attractor 22 Modes 300 determining 320, 331 N Navier-Stokes equations 98, 335 O Operator evolutionary 11 Frechét differentiable at a point 39 hyperbolic 370 monodromy 14, 397 potential 108 regular value 114 with discrete spectrum 77, 218 Orbit see Trajectory P Phase space 11 Point equilibrium 17 fixed 17 hyperbolic 39, 116, 268 heteroclinic 367 homoclinic 367 transversal 381 recurrent 49 stationary 17 Poisson stability 49 Process 322 Projector 55 R Radius of dissipativity 24 Reduction principle 49 S Scale of Hilbert spaces 79 Semigroup property 11 Semitrajectory 17 Set =-limit 18 M-limit 18 absorbing 24, 30 fractal 53 hyperbolic 377 inertial 61, 65, 67, 103, 254 418 Index

invariant 18 negatively invariant 18 of asymptotically determining functionals 286 of determining functionals 292 positively invariant 18 asymptotically stable 45 Lyapunov stable 45 uniformly asymptotically stable 45 unessential, with respect to measure 22 unstable 35 Sobolev space 93, 306 Solution mild 85 strong 82 weak 82, 223, 232 Stability asymptotic 45 uniform 45 Lyapunov 45 Poisson 49 Star-like domain 307 T Trajectory 17 heteroclinic 367 homoclinic 367 transversal 381, 402 induced 161 periodic 17 Poisson stable 49 U Upper semicontinuity of attractors 46 V Value of operator, regular 114 Volume averages, determining 311, 331