A Dissertation entitled

Sobolev Gradient Semi-flows & Applications to Nonlinear Problems

by Ramesh Karki

Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Mathematics

Dr. Alessandro Arsie, Committee Chair

Dr. Denis A. White, Committee Member

Dr. Franco Cardin, Committee Member

Dr. Gregory S. Spradlin, Committee Member

Dr. Mao-Pei Tsui, Committee Member

Dr. Patricia R. Komuniecki, Dean College of Graduate Studies

The University of Toledo May 2015 Copyright 2015, Ramesh Karki

This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. An Abstract of Sobolev Gradient Semi-flows & Applications to Nonlinear Problems by Ramesh Karki

Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Mathematics The University of Toledo May 2015

We are interested in solving nonlinear pseudo-differential equations (in particular, partial differential equations as well) involving fractional powers of uniformly ellip- tic self-adjoint operators of order two with suitable smoothness conditions on the coefficients subject to given (Dirichlet or periodic) boundary conditions. Under the stronger assumptions, we are interested in studying solutions in a special class whose elements satisfy non-selfintersecting property and have bounded distance from a given hyperplane, since such solutions are the analogue for Aubrey-Mather sets for ODEs and leaves of minimal foliations or laminations for PDEs . To solve such a ΨDE, we will start by introducing an energy type functional whose Euler-Lagrange equation is the pseudo-differential equation itself. As we seek to minimize this functional, we will introduce the Sobolev gradient of the functional as an element of a suitable Sobolev space and then we consider the gradient descent equation subject to appropriate ini- tial and boundary conditions. The equilibrium solutions of this Sobolev gradient descent equation are the critical points we are looking for.

Now the first step of our work will be to construct a semi-flow corresponding to the aforementioned initial-boundary value problem. So we will prove the existence, uniqueness, regularity, and comparison properties related to the semi-flow. Then

iii the next step will be to analyze the convergence of this semi-flow to an equilibrium solution to this initial-boundary value problem. In our work, we will adapt two methods: analytical method and numerical method. We apply various analytical tools to establish the general results and numerical tools to study concrete solutions of particular pseudo-differential or partial differential equations.

iv Dedicated to

“My Professors, Teachers, Family, and Friends” who have directly or indirectly supported me in my academic life Acknowledgments

I would like to sincerely thank my Ph. D. adviser Prof. Alessandro Arsie for his continuous advice and tireless support to develop foundation in my research area and successfully establish main results and many other technical results included in this dissertation. I would like to thank my dissertation committee members Dr. Denis

A. White, Dr. Mao-Pei Tsui, Dr. Franco Cardin and Dr. Gregory Spradlin for their careful reading of this manuscript, feedback and suggestions.

I would also like to thank Dr. Timothy Blass and Prof. Rafael de lLave for their support in many ways. Their kind responses of my questions in emails and in person during several meetings, their help on providing notes, papers and books, and giving comments were instrumental to proceed my research work ahead.

I am particularly indebted to my good friends and professors Dr. Akaki Tikaradze,

Dr. Vani Cheruvu, Dr. Santosh Kandel, Dr. Harihar Khanal, who have been help- ing me discuss on several topics and problems, giving me valuable suggestions and encouragements to obtain my academic achievements by several means.

Finally, I would like to express my sincere thanks to my family, many other friends, colleagues, and professors who have influenced directly and indirectly in my academic life so that their support has immensely helped me obtain my academic achievements.

Ramesh Karki

vi Contents

Abstract iii

Acknowledgments vi

Contents vii

List of Abbreviations x

List of Symbols xi

Preface xii

1 Introduction 1

2 Some Preliminaries 5

2.1 Euler-Lagrange Equation ...... 5

2.2 Sobolev Spaces ...... 7

2.2.1 When Ω = Td ...... 7 2.2.2 When Ω is an open bounded domain ...... 10

2.3 Sobolev Gradients ...... 11

2.4 Sobolev Gradient Descent Equations ...... 14

2.5 Semigroups And Fractional Powers of Linear Operators ...... 16

2.6 The Spectral Property For An Unbounded Self-adjoint Operator On A

Hilbert Space ...... 22

2.7 Main Results ...... 23

vii 2.7.1 Existence, Uniqueness & Regularity ...... 23

2.7.2 The Comparison Principles ...... 24

2.7.3 Applications To Semilinear ΨDEs ...... 25

2.7.4 Numerical Solutions To Some ΨDEs ...... 25

3 Characterization of Sobolev Spaces and Fractional Powers of Oper-

ators 27

3.1 Characterization of L2(Ω)...... 27

3.2 Characterizations of Hα(Ω), Hαβ(Ω) and Fractional Powers of Oper-

ators on L2(Ω)...... 28

3.3 Fractional Powers of Operators on Hαβ(Ω)...... 30

4 Constructions of Semigroups, Comparison, Regularity & Bounded-

ness 32

4.1 Constructions of Semigroups and Comparison ...... 33

4.2 Regularity and Boundedness of e−tAα ...... 37

4.3 Constructions of Semigroups and Comparison, revisited ...... 41

4.4 Regularity and Boundeness Properties of etL ...... 46

5 The Proofs of Main Results 50

5.1 The Proofs of Main Theorems Under L2-gradient ...... 50

5.1.1 Some Preliminaries ...... 50

5.1.2 The Proof of Theorem 2.7.1 ...... 54

5.1.3 The Proof of Theorem 2.7.3 ...... 58

5.2 The Proofs of Main Theorems Under Hαβ-gradient ...... 62

5.2.1 Some Preliminaries (Revisited) ...... 62

5.2.2 The Proof of Theorem 2.7.2 ...... 65

5.2.3 The Proof of Theorem 2.7.4 ...... 70

viii 6 Applications to Semilinear Pseudo Differential Equations 75

6.1 Nonself-intersecting Solutions ...... 76

6.2 A Generalization of Minimal Foliations (or Minimal Laminations) &

Invariant Tori (or Aubry-Mather Sets) ...... 77

6.3 The Proof of Theorem 2.7.5 ...... 82

6.4 Regularity in Some Special Cases ...... 91

6.5 Numerical Solutions to Some Semilinear PDEs and ΨDEs ...... 92

References 100

A Pseudo-differential Operators 103

B MATLAB Programmings 107

B.1 Code for Example 6.5.1 ...... 107

ix List of Abbreviations

E-L ...... Euler-Lagrange ACP ...... abstract Cauchy aroblem IVP ...... initial value problem IBVP ...... initial-boundary value problem ΨDE ...... pseudo-differential equation PDE ...... partial differential equation ODE ...... ordinary differetial equation FFT ...... fast Fourier transform DFT ...... discrete Fourier transform

x List of Symbols

R ...... The set of real numbers N ...... The set of positive integers NTd ...... The torus or N copies of [0,N]d or Rd/NZd

C∞(Ω)..... The space of smooth functions in Ω ∞ Cc (Ω)..... The space of smooth functions with compact supports in Ω S(R)...... The Schwarz space of rapidly decreasing smooth functions 0 ∞ D (Ω)...... The space of distributions or dual of Cc (Ω) S0(R)...... The space of tempered distributions or dual of S(R) L∞(Ω)...... The space of functions whose essentialsup norm is finite L2(Ω)...... The space of functions (more generally distributions) which are square integrable over Ω and satisfy either periodic or Dirichlet boundary conditions Hk(Ω)...... Sobolev space of order k consisting of distributions in L2(Ω) whose weak partial derivatives of orders ≤ k, k ∈ Z+ are in L2(Ω) Hs(Ω)...... Fractional order Sobolev space of order s Hαs(Ω)..... Fractional order Sobolev space of order αs

Inner Products( ., .)L2 = (., .)L2(Ω) , (., .)αβ = (., .)Hαβ (Ω) , (., .)αs = (., .)Hαs(Ω) , (., .)s = (., .)Hs(Ω) , (., .)k = (., .)Hk(Ω) , (., .)0 = (., .)H0(Ω)

Norms ...... k.k∞ = k.kL∞(Ω), k.kL2 = k.kL2(Ω), k.kαβ = k.kHαβ (Ω), k.kαs = k.kHαs(Ω), k.ks = k.kHs(Ω), k.kk = k.kHk(Ω), k.k0 = k.kH0(Ω)

L(H)...... The Banach space of all bounded linear operators on Hilbert space H under supnorm of operators.

α ∂|α| d D ...... = for multi-index α = (α1, . . . , αd) ∈ N with |α| = α1 + ∂xα1 ...∂xαd ∂αi th ··· + αd, where each is αi weak partial derivative. ∂xαi

α α1 αd d |x| ...... = x1 ...... xn for multi-index α = (α1, . . . , αd) ∈ N with |α| = α1 + ··· + αd.

xi Preface

The goal of this dissertation work is to give a brief presentation of Sobolev gradi- ents and to emphasize their applications on the theory of partial differential equations and pseudo-differential equations. We explore in depth the solutions to such equa- tions and some interesting behavior of the solutions using powerful tools from some of the rich theories of analysis such as the abstract semigroup theory, the spectral theory of linear unbounded self-adjoint operator on a Hilbert space, the theory of gen- eralized functions, functional analysis, and theory of partial differential equations and pseudo-differential equations. It, further, highlights on some numerical applications of these equations and opens up some problems in the future directions.

xii Chapter 1

Introduction

In this chapter we are going to discuss briefly what our main goal is, how we set

up problem, and what we do to solve the problem we set up. We also discuss about

what we prove so far, work in progress and the possible extensions of our problem or

some future projects.

Here our main interest is to solve certain types of nonlinear partial differential

equations (PDEs) and pseudo-differential equations (ΨDEs),

look for some special classes of solutions and investigate various properties. We

will investigate these equations and their solutions via analytical method as well as

numerical method.

To deal with each of these nonlinear PDEs and ΨDEs we consider in section 2.1 a

certain energy type functional Eα defined on an infinite dimensional Hilbert space H, namely a Sobolev space Hs(Ω), s ≥ 0, so that the equation satisfied by its critical points is the given PDE or ΨDE. We discuss these Sobolev spaces in section 2.2 and characterize them in sections 3.1 and 3.2. As the critical points of Eα are of our interest, we need to construct a suitable gradient of the functional as an element

of H. Such a gradient, denoted by ∇HEα(u), is a counterpart of the usual gradient

Pd ∂g 1 d ∇g = i=1 of a real valued C function g defined on the Euclidean space R . This ∂xi gradient needs to have the property that the solutions to the corresponding gradient

1 equation ∇HEα(u) = 0 are the critical points of the functional Eα. We consider H as a Sobolev space, introduce such a gradient as an element of the Sobolev space in section 2.3 and also give the formula for it there. Our method of solving a PDE or

ΨDE by finding critical points of the functional Eα is quite different from the method of solving a PDE by finding minimizers of an energy functional using the so called direct method of calculus of variations.

By the direct method of calculus of variations, a variational problem is solved by

finding minimizers of an energy functional. These minimizers with enough smoothness are the solutions to a PDE, which we call Euler-Lagrange equation for the energy functional. But unlike this method of finding the minimizers of Eα directly, we look for functions u which solve the gradient equation ∇HEα(u) = 0. We will show that this equation is in fact equivalent to the Euler-Lagrange equation of Eα. In section 2.2 we will introduce fractional order Sobolev spaces and continue them in Chapter 2 where we characterize them with the help of spectral theory of unbounded self-adjoint operator on a Hilbert space. In section 2.3 we define Sobolev gradients of Eα and give formulas for them.

We consider in section 2.4 the gradient descent equation

∂u = −∇ E (u) ∂t H α subject to the initial condition u(0,.) = u0(.) in appropriate spaces and the periodic or Dirichlet boundary conditions so that this initial-boundary value problem is well- posed. As we are seeking for finding functions u that solve the gradient equation

∇HEα(u) = 0, we need to look for equilibrium solutions to this initial-boundary value problem. To solve this IBVP, we recast it into the corresponding abstract

Cauchy problem (ACP)  du  = −∇HEα(u) dt (1.0.1)  u[0] = u0

2 on the Sobolev space whose elements are accompanied by either of the above boundary conditions. The main part of this thesis contributes for proving the existence and uniqueness of solution to this ACP, improving regularity of the solution, establishing the comparison principle related to the solution, and applying these results to study the convergence of the solution under special circumstances.

In section 2.5 and section 2.6 we briefly discuss about the semigroup theory, some important results from spectral theory of unbounded self-adjoint operators on a Hilbert space and discuss the fractional powers of the self-adjoint operators (also in section 3.3). In section 2.7 we state some of the main original results we proved in this thesis such as the existence, uniqueness, regularity and comparison principle for the solution to the above ACP and the existence of solutions to the Euler-Lagrange equation of Eα in a class Bω. We will briefly introduce this class in this chapter and discuss about it later in Chapter 6.

Since the proof of the main results rely heavily on semigroup theory, throughout

Chapter 4 we construct various semigroups generated by a self-adjoint uniformly elliptic operator and its fractional powers on Sobolev spaces and discuss comparison, regularity and boundedness properties of these semigroups.

In Chapter 5 we first define a mild solution to ACP (1.0.1) as an integral form of the ACP using the semigroups constructed in Chapter 4. We prove that this solution is a globally attracting fixed point of a contraction map by using Banach

Fixed Point Theorem. Then we improve its regularity and complete the proof of existence, uniqueness and regularity results. Next we adapt the iteration method to prove the comparison principle by proving that these iterations satisfy the comparison principle and converge to the corresponding fixed points.

In Chapter 6 as another original main result of this dissertation, we prove the existence of solutions to the Euler Lagrange equation of Eα in the class Bω when ω is

d d a vector in Q . The class Bω consists of functions u : R → R with the properties that

3 the graph of each such function when projected on a torus does not intersect itself

(nonself-intersecting property) and that the hypersurface represented by xd+1 = u(x) has bounded distance from the hyperplane xd+1 = ω.x. We use the main results- the existence and uniqueness theorem, the regularity theorem and the comparison prin- ciple mentioned above to prove that the solution u(t, x) to the IBVP corresponding to ACP (1.0.1) starting at a point in Bω always remains in Bω. Moreover, for such a solution u(t, x) there is an increasing sequence of times {tn} approaching ∞ such that u(tn, x) converges to uω(x) as tn approaches ∞ and uω satisfies the Euler-Lagrange equation. But since Bω is closed under this convergence, uω is in Bω. In this way we prove the existence of solutions to the Euler-Lagrange equation of Eα in the class Bω.

The work of investigating whether the analogous result holds when ω is in Rd − Qd is in progress.

Our interest for studying the solutions to semilinear pseudo-differential equations

(Euler-Lagrange equations for Eα) in the class Bω stems from the study of minimal fo- liations or laminations for variational problems for PDEs and study of Aubrey-Mather sets for ODEs. Such a minimal foliation for a variational problem for PDE is a con- tinuous one parameter family of smooth hypersurfaces as graphs of functions, called leaves of the foliation, covering the fundamental domain and the function representing each leaf satisfies nonself-intersecting property and is a solution to the correspond- ing Euler-Lagrange equation. Aubrey-Mather sets arise when studying the stability problem for dynamical systems, especially for the systems of two degree of freedom for which such a problem is about constructing two dimensional invariant tori (a part of KAM theory). The solutions to ΨDE in the class Bω have similar properties as compared with the leaves of minimal foliations. We will briefly discuss these minimal foliations or laminations (foliations with gaps) and Aubrey-Mather sets in Chapter 6.

To know more about the minimal foliations or laminations, one can read [3, 11, 12, 13], and to know more about Aubrey-Mather sets, one can read [14].

4 Chapter 2

Some Preliminaries

In the course of proving our original results we apply various tools from abstract semigroup theory on a Hilbert space, spectral theory of linear unbounded self-adjoint operator on a Hilbert space, and many other concepts from functional analysis, dis- tribution theory, and theory of partial differential equations and pseudo-differential equations.

Now we are going to discuss briefly some topics and key concepts which we fre- quently use in studying our original results in this entire dissertation.

2.1 Euler-Lagrange Equation

Let Ω be an open bounded domain in Euclidean space Rd with smooth boundary

∂Ω or a closed manifold such as a torus NTd. Let f(x, y) be a bounded continuous real-valued function on Ω¯ × R such that its continuous first partial derivative with ¯ respect to y, fy, exists in R for all x ∈ Ω and is bounded on Ω × R. We are interested in finding solutions of the following semi-linear ΨDE

Aαu + f(x, u) = 0, x ∈ Ω (2.1.1)

subject to Dirichlet boundary conditions u|∂Ω = 0 or periodic boundary conditions

5 u(x + ej) = u(x) ∀ ej = (0,..., 1,..., 0), j = 1, . . . , d, where α ∈ (0, 1], and A is a linear, self-adjoint and uniformly elliptic operator of order 2 on L2(Ω), the space of square integrable functions satisfying either of the boundary conditions, given by

d X (Au)(x) = − ((aij(x)uxi (x))xj , x ∈ Ω (2.1.2) i,j=1

∞ with the properties that aij ≡ aji ∀ i, j = 1, 2 . . . , d and aij ∈ C (Ω), and

d 2 X 2 d θ1|ξ| ≤ aij(x)ξiξj ≤ θ2|ξ| ∀ x ∈ Ω, ∀ ξ ∈ R . (2.1.3) i,j=1

for some positive real numbers θ1, θ2.

The semi-linear ΨDE (2.1) is the Euler-Lagrange equation of the functional Eα,

Z 1 α/2 2 Eα(u) = |A u| + F (x, u) dx, α ∈ (0, 1] (2.1.4) Ω 2

R y ∞ defined on a Sobolev space, where F (x, y) = 0 f(x, z) dz. To see this, if φ ∈ Cc (Ω) then

d E (u + τφ)| dτ α τ=0 Z d n1 α/2 α/2 o = (A (u + τφ),A (u + τφ))L2(Ω) + F (x, u + τφ) dx |τ=0 dτ 2 Ω Z α/2 α/2 ∂ α/2 =(A u, A φ)L2(Ω) + F (x, u + τφ)|τ=0 dx (∵ A is linear) Ω ∂τ Z α α/2 =(A u, φ)L2(Ω) + f(x, u)φ dx (∵ A is self- adjoint) Ω α =(A u + f(x, u), φ)L2(Ω) = 0,

α ∞ 2 implies that A u + f(x, u) = 0 by the denseness of Cc (Ω) in L (Ω).

We will see in section 2.4 that (2.1) is equivalent to each of ∇αβEα(u) = 0 and

∇L2 Eα(u) = 0 for each critical point u of Eα, where ∇αβEα(u) and ∇L2 Eα(u) are

6 αβ 2 H (Ω)-gradient and L (Ω)-gradient of Eα at u respectively. This is the reason why we focus on studying the solutions to these gradient equations in order to solve (2.1).

2.2 Sobolev Spaces

Following [2, 9, 12, 17, 22, 24], first we recall definitions of some fractional order

Sobolev spaces and then introduce new Sobolev spaces.

We start with recalling the following Sobolev spaces as a Hilbert space with the given inner product:

k 2 α 2 H (Ω) = {u ∈ L (Ω): D u ∈ L (Ω) ∀ |α| ≤ k}, k ∈ N,

where Ω is open bounded domain or a torus or all of Rd and Dα is a partial derivative of order α in the distributional sense, with inner product and norm

1/2 X α ν n X α 2 o (u, v)k = (D u, D v)L2 , kukk = kD ukL2 and |α|≤k |α|≤k

s d 0 d 2 s/2 2 d H (R ) = {u ∈ S (R ) : (1 + |ξ| ) uˆ ∈ L (R )}, s ∈ R,

with inner product and norm

2 s/2 2 s/2 2 s/2 (u, v)Hs( d) = ((1 + |ξ| ) u,ˆ (1 + |ξ| ) vˆ)L2( d), kuk s d = (1 + |ξ| ) uˆ 2 d . R R H (R ) L (R )

2.2.1 When Ω = Td

In this case, we always consider L2(Ω) together with the usual L2-inner product

as the space of all square integrable functions that satisfy the periodic boundary

conditions.

7 We discuss various fractional order Sobolev spaces which are Hilbert spaces with corresponding inner products (see [1]).

s 0 X 2 s/2 ij.x 2 s H (Ω) = {u ∈ D (Ω): (1 + |j| ) uˆ(j)e ∈ L (Ω)} = D(Λ ), s ∈ R j∈2πZd where Λsu(x) := P (1 + |j|2)s/2uˆ(j)eij.x for u ∈ D0(Ω), with inner product and j∈2πZd norm

s s X 2 s X 2 s 2 1/2 (u, v)s = (Λ u, Λ v)L2 = (1+|j| ) uˆ(j)vˆ(j), kuks = { (1+|j| ) |uˆ(j)| } j∈2πZd j∈2πZd

ij.x ik.x
since e , e L2 = δjk. In particular, if s = 0 then

X H0(Ω) = {u ∈ D0(Ω): uˆ(j)eij.x ∈ L2(Ω)} = L2(Ω)

j∈2πZd with inner product and norm

X X 2 1/2 (u, v)0 = uˆ(j)vˆ(j) = (u, v)L2 , kukL2 = { |uˆ(j)| } . j∈2πZd j∈2πZd

Moreover, if s ≥ 0 then P |uˆ(j)|2 ≤ P (1 + |j|2)s|uˆ(j)|2 ⇒ Hs(Ω) ⊂ j∈2πZd j∈2πZd L2(Ω). In general, it is clear that Hs(Ω) ⊂ Ht(Ω) for t ≤ s. The inclusion is continuous.

Notice that Λα = (I − ∆)α/2 on D0(Ω) so that D((I − ∆)α/2) = Hα(Ω), where the inner product on D((I − ∆)α/2) is the one which induces the graph norm. Thus now on, unless otherwise stated, we will consider the domain of a differential or pseudo- differential operator on L2(Ω) or on a Sobolev space with the inner product that induces the graph norm in order to make such a domain a new Sobolev space. For example, with the corresponding graph norms, we have D(I+A) = H2(Ω) = D(I−∆)

8 where A is as in (2.1.2).

We recall an intermediate space [Hr(Ω), Hs(Ω)] between Hr(Ω) and Hs(Ω) as

[Hr(Ω), Hs(Ω)] = H(1−θ)r+θs(Ω) for 0 ≤ θ ≤ 1 and r < s. Then Hs(Ω) ⊂

H(1−θ)r+θs(Ω) ⊂ Hr(Ω) and the norms satisfy the following interpolation inequal- ity

1−θ θ s kuk(1−θ)r+θs ≤ C kukr kuks for u ∈ H (Ω).

Since D(I+A) = H2(Ω) = D(I−∆), applying interpolation method (with proposition

2.2 in [9]), we get Hα(Ω) = D((I −∆)α/2) = D((I −∆)0(1−α/2)+1.α/2) = [L2(Ω),D(I −

2 α/2 ∆)]α/2 = [L (Ω),D(I + A)]α/2 = D((I + A) ). At this stage, we state the following lemma and wait until section 2.2 to prove it.

Lemma 2.2.1. The domain D((I + A)α/2) together with the inner product which

α/2 induce the graph norm kukD(I+A)α/2 = (I + A) u L2 and the domain D((I + α 1/2 A ) ) together with the inner product which induce the graph norm kukD(I+Aα)1/2 =

α 1/2 (I + A ) u L2 are equal.

This lemma and discussions preceding it allow us to define

Hα(Ω) = {u ∈ L2(Ω):(I + Aα)1/2u ∈ L2(Ω)} as the domain of (I + Aα)1/2 with inner product and norm

α 1/2 α 1/2
α 1/2 (u, v)α = (I + A ) u, (I + A ) v L2 , kukα = (I + A ) u L2 . and hence for β ∈ [0, 1]

Hαβ(Ω) = {u ∈ L2(Ω):(I + Aα)β/2u ∈ L2(Ω)}

9 with inner product and norm

α β/2 α β/2
α β/2 (u, v)αβ = (I + A ) u, (I + A ) v) L2 , kukαβ = (I + A ) u L2 .

Here the last Sobolev space once again follows from the interpolation method together

α with proposition 2.2 in [9] and the fact that Aα = I+A is a sectorial positive operator

2 2 on L (Ω) so that the fractional power of Aα on L (Ω) can be defined. For instance, the operator Aα satisfies the properties that ρ(Aα) contains the sector S0,φ = {0 6=

−1 λ ∈ C : φ ≤ | arg(λ)| ≤ π, φ ∈ (0, π/2)} and k(λ − Aα) kL(L2(Ω)) ≤ M/|λ| for

λ ∈ S0,φ for some M > 0 (for more detail, see section 1.3 in [26]).

2.2.2 When Ω is an open bounded domain

In this case, we always consider L2(Ω) together with the usual L2-inner product as the space of all square integrable functions that satisfy the Dirichlet boundary conditions.

As in the previous case, we consider the following Sobolev space

Hα(Ω) = {u ∈ L2(Ω):(I + Aα)u ∈ L2(Ω)} with inner product and norm

α α α (u, v)α = ((I + A )u, (I + A )v)L2 , kukα = k(I + A )ukL2 , and apply the interpolation method to obtain the following intermediate space be- tween L2(Ω) and Hα(Ω)

Hαβ(Ω) = {u ∈ L2(Ω):(I + Aα)β/2u ∈ L2(Ω)}, 0 < β < 1

10 with inner product and norm

α β/2 α β/2
α β/2 (u, v)αβ = (I + A ) u, (I + A ) v L2 , kukαβ = (I + A ) u L2 .

2.3 Sobolev Gradients

αβ αβ Sobolev gradient, namely H (Ω)-gradient of Eα at u ∈ H (Ω), is a unique element g ∈ Hαβ(Ω) such that

αβ DEα(u)η = (g, η)αβ ∀ η ∈ H (Ω),

where DEα(u) is the Fr´echet derivative of Eα at u. The Riesz Representation Theorem

αβ guarantees the existence of such a g ∈ H (Ω) since it is known that DEα(u) is a

αβ continuous linear functional on H (Ω). We denote g by ∇Hαβ (Ω)Eα(u) or simply

∇αβEα(u).

αβ Thus g ∈ H (Ω) is such that g = ∇αβEα(u) if and only if DEα can be expressed as

α β/2 α β/2
αβ DEα(u)η = (I + A ) g, (I + A ) η L2 ∀ η ∈ H (Ω).

0 2 In particular, if β = 0 then DEα(u)η = (g, η)0 = (g, η)L2 ∀ η ∈ H (Ω) = L (Ω) and

0 2 hence g = ∇0Eα(u) = ∇L2 Eα(u), H (Ω) or L (Ω)-gradient of Eα at u. Now we establish the formulas for Hαβ(Ω)-gradient and L2(Ω)-gradient.

∞ ∞ d Lemma 2.3.1. For every η ∈ Cc (Ω) or η ∈ C (Ω) when Ω = T , we have

α a) DEα(u)η = (A u + f(., u), η)L2 ,

α 1−β α −β
b) DEα(u)η = (I + A ) u − (I + A ) (u − f(., u)), η αβ

11 Proof. By definition of Fr´echet derivative, we have

|E (u + η) − E (u) − DE (u)η| lim α α α = 0, kηk →0 L2 kηkL2 from which it follows that

2 Eα(u + η) = Eα(u) + DEα(u)η + o(|η| ).

On the other hand

Z 1 α/2 2 Eα(u + η) = |A (u + η)| + F (., u + η) Ω 2 Z 1 α/2 α/2 2 2 = |A u + A η| + F (., u) + Fy(., u)η + o(|η| ) Ω 2 Z α 2 =Eα(u) + (η, A u)L2 + f(., u)η + o(|η| ) Ω α 2 =Eα(u) + (η, A u + f(., u))L2 + o(|η| ).

Since F is continuously twice differentiable with respect to y, in the second line we have used Taylor’s Theorem to write

1 F (., u + η) =F (., u) + F (., u)η + F (., u˜)η2 y 2! yy 1 =F (., u) + f(., u)η + f (., u˜)η2 2! y for someu ˜ between u and u + η.

Combining the above two expressions for Eα(u + η), we obtain part (a) of lemma.

12 Since (I + Aα)β/2 is a self-adjoint operator on L2(Ω), we have

α DEα(u)η = (A u + f(., u), η)L2

α β α −β α
= (I + A ) (I + A ) (A u + u − u + f(., u)), η L2

α β/2 α −β α α β/2
= (I + A ) (I + A ) (A u + u − u + f(., u), (I + A ) η L2

α −β α
= (I + A ) [(I + A )u − {u − f(., u)}], η αβ

α 1−β α −β
= (I + A ) u − (I + A ) {u − f(., u)}, η αβ .

Thus part (b) of lemma follows.

∞ 2 Since Cc (Ω) is dense in each of L (Ω), part (a) of Lemma 2.3.1 holds true for all

2 η ∈ L (Ω). Therefore, the third step of the calculation of DEα(u)η in the proof of part (b) of the lemma holds true for all η ∈ L2(Ω) such that (I + Aα)β/2η ∈ L2(Ω), that is,

α β/2 α −β α α β/2
DEα(u)η = (I + A ) (I + A ) (A u + u − u + f(., u), (I + A ) η L2 for all η ∈ D((I + Aα)β/2) = Hαβ(Ω). This implies that part (b) of lemma holds true for all η ∈ Hαβ(Ω). Hence we can conclude that

α ∇L2 Eα(u) = A u + f(., u) and (2.3.1)

α 1−β α −β ∇αβEα(u) = (I + A ) u − (I + A ) {u − f(., u)}. (2.3.2)

13 2.4 Sobolev Gradient Descent Equations

As we discussed in section 2.1.1, we want to solve nonlinear pseudo-differential equation (2.1) by finding the critical points of the functional Eα. Of these critical points, we may expect to find a minimizer of Eα. To find such a minimizer, we consider Hαβ(Ω)-gradient descent equation

∂u(t, x) = −∇ E (u(t, x)) ∂t αβ α which then by (2.3.2) becomes

∂u(t, x) = −(I + Aα)1−βu(t, x) + (I + Aα)−β(u − f(x, u(t, x))). ∂t

In this equation, we set Lu = −(I + Aα)1−βu as a linear operator L and Nu = (I +

α −β A ) (u−f(x, u)) as a nonlinear operator N, impose initial condition u(0, x) = u0(x) and impose either of Dirichlet boundary conditions u(t, .)|∂Ω = 0 or the periodic boundary conditions u(t, x + Nej) = u(t, x), t ≥ 0 for all ej = (0,..., 1 ..., 0), j = 1, 2 . . . d, x ∈ Ω depending on whether Ω is an open bounded domain with smooth boundary or a torus NTd, where N ∈ N. We express u(t, x) = u[t](x) by fixing x to represent t 7→ u[t] as a map from time interval into a function space and consider L2(Ω) as the space of square integrable functions over Ω that satisfy either of above boundary conditions so that the functions in Hαs(Ω), s ≥ 0 also accompany either boundary condition. Then the above initial- boundary value problem (IBVP) can be expressed into the abstract Cauchy problem

(ACP)  du[t]  = L(u[t]) + N(u[t]) if t > 0 dt (2.4.1)  u[0] = u0.

14 If we replace Hαβ(Ω)-gradient by L2(Ω)-gradient in the above discussion, we obtain, in particular, the ACP

 du[t] α  = −A u − f(., u) if t > 0 dt (2.4.2)  u[0] = u0.

We notice that the equilibrium solutions to (2.4.1) satisfy −∇αβEα(u) = 0 which is equivalent to −(I +Aα)u+I(u−f(., u)) = 0 and thus equivalent to the semi-linear pseudo-differential equation (2.1), the Euler-Lagrange equation of Eα. Similarly the equilibrium solutions to (2.4.2) also satisfy (2.1). Therefore, the existence of a unique solution u(t, x) to each of last two ACPs (2.4.1) and (2.4.2) is a key in studying such equilibrium solutions that solve (2.1) and hence in studying the critical points of Eα. By the Hαβ(Ω)-gradient method, we mean the method of finding the critical

2 points of Eα by finding the equilibrium solutions to (2.4.1) and by the L (Ω)-gradient method, we mean the method of finding the critical points of Eα by finding the

αβ equilibrium solutions to (2.4.2). Although the formula for the H (Ω)-gradient of Eα appears to be more complicated than that of its L2(Ω)-gradient, we will equally focus on both the methods to solve (2.1.1) analytically. On the other hand, we prefer to

αβ use the H (Ω)-gradient method to find the critical points of Eα due to the reason given below.

In order to deal with the concrete solutions to some nonlinear ΨDEs of type

(2.4.1) and (2.4.2) and their convergence, we need to rely, in many situations, on the numerical solutions of them. For this, we need to choose an efficient method to perform the numerical simulations. The reason behind this is: the numerical solution to (2.4.1) in higher Fourier frequency mode converges to its equilibrium solution much faster than that of (2.4.2). In this perspective, the use of the Hαβ(Ω)-gradient is more efficient than the use of the L2(Ω)-gradient in terms of cost of time spent on numerical

15 computations to find equilibrium solutions to (2.4.1) and (2.4.2).

2.5 Semigroups And Fractional Powers of Linear

Operators

A semigroup of linear operators on a Banach space, particularly on a Hilbert

space in our case, is one of the strong tools to study the abstract theory of PDEs and

ΨDEs. The development of proofs of our main results in our work heavily rely on

semigroup theory of linear operators. In our work we will construct various semigroups

generated by closed densely defined unbounded selfadjoint operators on a Hilbert

space, namely by −A and negative of fractional powers of operators involving A on

Sobolev spaces. We apply these semigroups and the spectral theory of an unbounded

selfadjoint operator on a Hilbert space to define various fractional powers of the

operator and establish various interesting properties.

Following [8, 16, 23, 22]) we present some definitions and important properties

below.

Definition 2.5.1. A semigroup of bounded linear operators on a Hilbert space H is a one parameter family of bounded linear operators T (t), t ≥ 0 on H satisfying T (0) = I and T (t1 + t2) = T (t1) ◦ T (t2) ∀ t1, t2 ≥ 0. It is a strongly continuous semigroup or

C0-semigroup of bounded linear operators on H if it satisfies an additional property: for each u ∈ H, limh↓0 kT (h)u − ukH = 0.

Further a C0-semigroup of bounded linear operators on H is a C0 semigroup of contractions if kT (t)kL(H) ≤ 1 for t ≥ 0.

The elliptic operator A and its fractional powers are unbounded linear operators on Sobolev spaces, but we are going to use semigroups of bounded linear operators as main tools in our study. Therefore, it is necessary to connect these operators with

16 semigroups of bounded linear operators. The following definition will open the door

to enter in this direction.

Definition 2.5.2. Given a semigroup of bounded linear operators T (t), t ≥ 0 on a

Hilbert space H, a linear operator B defined on H by

T (h) − I D(B) = {u ∈ H : lim u exists. } and h↓0 h

T (h) − I d+T (t) Bu = lim u = u|t=0 for u ∈ D(B) h↓0 h dt

is called an infinitesimal generator of the semigroup T (t), t ≥ 0.

It is known ( see Theorem 1.2.6 in [16]) that B can not generate more than one

semigroup on H. We prefer to adapt commonly used notation etB for each T (t)

to emphasize the presence of the generator and simply say that B generates the

semigroup etB, t ≥ 0 on H. Here etB is merely a notation, but if B were a bounded

tB tB P∞ tnAn linear operator then e would be a bounded linear operator given by e = n=0 n!

tB tkBkL(H) with e L(H) ≤ e .

A C0-semigroup generated by an operator on a Hilbert space enjoys many inter- esting properties. We mention some of them below without proof (for proofs one can

see [16, 23]).

tB Suppose e , t ≥ 0 is a C0-semigroup of bounded linear operators generated by B on H.

tB SG1. For each u ∈ H, the (orbit) map ξu : t 7→ e u is continuous from [0, ∞) into

H. It is differentiable on (0, ∞) if and only if ξu is differentiable at t = 0.

SG2. For each u ∈ H and t > 0,

1 Z t+h lim esBu ds = etBu. h↓0 t t

17 SG3. For each u ∈ D(B), etBu ∈ D(B) and

d etBu = etBBu = BetBu for t > 0, dt

d+ etBu = Bu for t = 0, dt

R t sB SG4. For each u ∈ H and t ≥ 0, 0 e u ds ∈ D(B) and

Z t etBu − u =B esBu ds 0 Z t = esBBu ds if u ∈ D(B). 0

SG5. There exist M ≥ 1 and ω ≥ 0 such that

tB ωt e L(H) ≤ Me ∀ t ≥ 0.

provided that etB, t ≥ 0 is not necessarily a contraction semigroup.

SG6. B is a closed and densely defined linear operator that determines the semigroup

etB, t ≥ 0 uniquely.

tB From the above properties of semigroup, if e , t ≥ 0 is a C0-semigroup of con-

tB tractions on H and u ∈ D(B) then we can observe that the orbit map ξu : u 7→ e u

1 belongs to C([0, ∞),D(B)) ∩ C ((0, ∞),H). For, if t0 ≥ 0 then for any t > 0, as a consequence of SG4, we have

Z t tB t0B sB e u − e u = e Bu ds for t ≥ t0, t0

Z t0 tB t0B sB e u − e u = − e Bu ds for t < t0, t

18 from which we obtain that

tB t0B e u − e u H ≤ kBukH |t − t0|

This shows that ξu ∈ C([0, ∞),D(B)) provided the norm on D(B) induced from the

norm on H. The differentiability of ξu on (0, ∞) follows from SG1 and SG3. To

0 show ξu is continuous on (0, ∞), let us take t0 ≥ 0 and a sequence {tn} in (0, ∞)

tnB t0B with tn → t0. Then continuity of ξBu implies that e Bu → e Bu. By SG3,

tnB t0B 0 Be u → Be u showing the continuity of ξu at t0.

Thus property SG1 tells us that we can express D(B) = {u ∈ H : ξu is differentiable} ˙ and Bu = ξu(0). Property SG3 together with the last observation plays a vital role

tB in our work. The key point is that for every u0 ∈ D(B), u(t) = e u0 = ξu0 (t) is a solution to the abstract Cauchy problem

 du  = Bu, t > 0  dt  u(0) = u0.

Moreover every solution starting at a point in D(B) always remains in D(B). By

property SG6 the solution u(t) is unique.

The following theorems characterize the (infinitesimal) generator of a C0-semigroup of bounded linear operators on H.

Theorem 2.5.3 (Hille-Yosida). A densely defined closed linear operator B, not nec-

essarily bounded, on H is the infinitesimal generator of a C0-semigroup T (t), t ≥ 0

ωt of bounded linear operators with kT (t)kL(H) ≤ e for some ω ≥ 0 if and only if the resolvent set ρ(B) contains the interval (ω, ∞) and for every λ ∈ (ω, ∞),

−1 1 (λI − B) ≤ . L(H) λ − ω

19 In particular, it follows form Hille-Yosida Theorem that a densely defined closed linear operator B on H generates a C0-semigroup of contractions if and only if ρ(B) ⊃ (0, ∞) and for every λ ∈ (0, ∞),

−1 1 (λI − B) ≤ . L(H) λ

Definition 2.5.4. Suppose B is a linear operator on H. Then we say that B is maximal dissipative or simply m-dissipative if (−Bu, u) ≥ 0 for all u ∈ D(B) and

R(λ0 − B) = H for some λ0.

Theorem 2.5.5 (Lumer-Philips). Suppose B is densely defined closed linear opera- tor on H. Then B is an infinitesimal generator of C0-semigroup of bounded linear operators on H if and only if B is m-dissipative.

Theorem 2.5.6 (Representation formula for resolvent). Let B be the infinitesimal generator of a C0-semigroup on H. For each λ with Re(λ) > ω (ω as in SG5 above)

Z ∞ R(λ : B)u := (λI − B)−1u = e−λtetBudt for u ∈ H. 0

It seems from the last theorem that the resolvent of B is the Laplace transform of the semigroup. The inversion of this Laplace transform gives a representation formula for the semigroup (see Theorem 1.7.4, Corollary 1.7.5 and Theorem 1.7.7 in [16]). But here we like to include a couple of other representation formulas for the semigroup.

Theorem 2.5.7 (Representation formula for semigroup). Let B be the infinitesimal generator of a C0-semigroup of contractions on H and Bλ be the Yosida approximation

2 of B given by Bλ = λBR(λ : B) = λ R(λ : B) − λI. Then

etBu = lim etBλ u for u ∈ H. λ→∞

20 Theorem 2.5.8 (The exponential formula for semigroup). Let B be the infinitesimal

generator of a C0-semigroup on H. Then

t etBu = lim (I − B)−nu for u ∈ H. n→∞ n

Suppose a densely defined closed linear operator B on H is an infinitesimal gen-

tB erator of a C0 semigroup of contractions e , t ≥ 0. Then −B is a closed positive operator with D(B) dense in H (ref [10]). Referring to section 11 of chapter IX in

[20], we can obtain the following expressions for the fractional powers of −B

sin απ Z ∞ (−B)αu = sα−1(sI − B)−1(−B)u ds, u ∈ D(B), (2.5.1) π 0

1 Z ∞ (−B)αu = s−α−1(esB − I)u ds, u ∈ D(B) (2.5.2) Γ(−α) 0

α and a C0-semigroup of contractions generated by −(−B) on H

  R ∞ sB α  ft,α(s)e u ds, if t > 0 e−t(−B) u = 0 (2.5.3)  u, if t = 0

where ft,α in the integrand defined by

  1 R σ+i∞ zs−tzα  2πi σ−i∞ e dz, if s ≥ 0 ft,α(s) = (2.5.4)  0, if s < 0 with σ > 0, t > 0 and 0 < α < 1 is nonnegative for all s > 0.

As we sum up we have the following:

Theorem 2.5.9. (−B)α given by (2.5.1) or (2.5.2) is an infinitesimal generator of

−t(−B)α a C0-semigroup of contractions e , t ≥ 0 on H given by (2.5.3).

Finally, we will introduce definitions and properties of some other types of semi- 21 groups such as analytic semigroup and compact semigroup whenever we need them.

2.6 The Spectral Property For An Unbounded Self-

adjoint Operator On A Hilbert Space

In this section, we will briefly discuss spectral integral from spectral measure

theory of an unbounded self-adjoint operator on a Hilbert space. The goal of such

integral is to define fractional powers of a positive definite self-adjoint operators on

Sobolev spaces involving A and to characterize Sobolev spaces we discussed in section

2.2. These will help us in constructing semigroups of bounded linear operators and

in opening ways to establish other interesting properties.

Let us recall from [24] that a spectral measure E maps elements of σ-algebra A of subsets of a measurable subset of R to orthogonal projections of a Hilbert space H satisfying monotonicity and countably additivity conditions. If EB is spectral measure supported on the spectrum σ(B) of a positive unbounded self-adjoint operator B on

H and f is a Borel measurable EB-finite a.e. function on [0, ∞), that is EB({x ∈ [0, ∞): f(x) = ∞}) = 0, then the spectral integral for f(B) is given by

  R ∞ 2 D(f(B)) = {u ∈ H : 0 |f(λ)| dhEB(λ)u, ui < ∞}, (2.6.1)  R ∞ f(B)u = 0 f(λ) dhEB(λ)u, ui

In particular, for any α ∈ (0, 1) the operator Bα : H → H is defined by

  α R ∞ α 2 D(B ) = {u ∈ H : 0 |λ | dhEB(λ)u, ui < ∞}, (2.6.2)  α R ∞ α B u = 0 λ dhEB(λ)u, ui

Moreover if λ0 ∈ σ(B), then EB({λ0}) is orthogonal projection of H on eigenspace

22 Eλ0 corresponding to λ0.

2.7 Main Results

In this section, we state our original main results that we are going establish

throughout our work.

2.7.1 Existence, Uniqueness & Regularity

We know that L∞(Ω) ⊆ L2(Ω). The boundedness of f implies that the nonlinear

term f(., u) belongs to L2(Ω). So we can expect the regularity of solution to ACP

(2.4.2) to be of order 2α because the order of the linear pseudo-differential operator

Aα is 2α. We will prove in section 5.2 that the regularity of the nonlinear term N(u)

in ACP (2.4.1) has regularity 2αβ. But the order of the linear pseudo-differential

operator L in (2.4.1) is 2α(1 − β). Therefore, we can expect the regualrity of solution

to (2.4.1) to be of the order 2α(1 − β) + 2αβ = 2α again. Intuitively, it makes sense of having the regularity of solutions to both equations (2.4.2) and (2.4.1) the same because their corresponding equilibrium solutions are going to satisfy the same equation (2.1.1).

The first couple of results below are related to the existence and uniqueness of solution to each of (2.4.1) and (2.4.2).

Theorem 2.7.1. Let f be a bounded continuous function on Ω¯ × R such that the

1 map y 7→ f(x, y) is in C (R, R) for all x ∈ Ω and fy is bounded for all x ∈ Ω. If

∞ α ∞ u0 ∈ L (Ω) ∩ D(A ) then there exists a unique solution u in C([0, ∞), L (Ω) ∩ H2α(Ω)) ∩ C1((0, ∞), L∞(Ω) ∩ L2(Ω)) to the ACP (2.4.2).

∞ Theorem 2.7.2. Let f be as mentioned in Theorem 2.7.1. If u0 ∈ L (Ω)∩D(L) then there exists a unique solution u in C([0, ∞), L∞(Ω) ∩ H2α(Ω)) ∩ C1((0, ∞), L∞(Ω) ∩

H2αβ(Ω)) to the ACP (2.4.1). 23 From the results of last two theorems, it follows that there exists a unique solution to each of the following IBVPs:

  ∂ α  u(t, x) = −A u(t, x) − f(x, u(t, x)) for t > 0 ∂t (2.7.1)  u(0, x) = u0(x) and

 ∂  u(t, x) = −(I + Aα)1−βu(t, x) + (I + Aα)−β{u(t, x) − f(x, u(t, x))} for t > 0 ∂t  u(0, x) = u0(x) (2.7.2) for almost every x ∈ Ω, subject to either boundary conditions, Dirichlet or peri- odic. Here the equalities in either equation have to be understood in the sense of corresponding Sobolve spaces.

2.7.2 The Comparison Principles

Next couple of results are comparison principles for the solutions to (2.4.1) and

(2.4.2).

Theorem 2.7.3. Suppose f is as in Theorem 2.7.1 and is nonincreasing in y variable.

Let u and v be two solutions of initial-boundary value problem (2.7.1) corresponding

∞ α to the initial data u0 and v0 respectively, where u0, v0 ∈ L (Ω) ∩ D(A ). If u0 ≥ v0 almost every x ∈ Ω, then u(t, x) ≥ v(t, x) almost every x ∈ Ω and for all t ∈ [0, ∞).

Theorem 2.7.4. Suppose f is as in Theorem 2.7.1 and is nonincreasing in y variable.

Let u and v be two solutions of initial-boundary value problem (2.7.2) corresponding

∞ to the initial data u0 and v0 respectively, where u0, v0 ∈ L (Ω) ∩ D(L). If u0 ≥ v0 almost every x ∈ Ω, then u(t, x) ≥ v(t, x) almost every x ∈ Ω and for all t ∈ [0, ∞).

24 2.7.3 Applications To Semilinear ΨDEs

Under suitable periodicity conditions on the nonlinear function f and coefficients functions aij of the linear self-adjoint elliptic operator A, we first reduce our varia-

tional type problem, a problem of finding critical points of Eα, on a torus. Then we

find solutions in a class Bω to the periodic boundary value problem (2.1.1), where

d each element u in Bω provided ω ∈ Q satisfies the non-selfintersecting property and

the hypersurface xd+1 = u(x) determined by u has a bounded distance from a fixed hyperplane xd+1 = ω.x (see Chapter 6 for more detail).

d+1 ∞ d Theorem 2.7.5. Let f ∈ C(T , R) with fy continuous and aij ∈ C (T , R). Then

d α for every ω ∈ Q , there exists u ∈ Bω satisfying A u + f(., u) = 0.

2.7.4 Numerical Solutions To Some ΨDEs

Finally as numerical applications, we develop MATLAB programming for solving

some semilinear partial differential equations and pseudo-differential equations of type

(2.1.1) subject to periodic boundary conditions.

For example, we solve the 2D-semilinear boundary value problem

(−∆)αv + 2π sin(2π(x + y)) cos(2πv) = 0

2 subject to v(x + ei) = v(x), where ei, i = 1, 2 are the standard unit vectors in R . Indeed, we develop the MATLAB programming based on the corresponding gra-

dient descent equations subject to suitable initial condition and periodic boundary

conditions using discrete Fourier transform. Then from the numerical simulation we

obtain solution to this IBVP which remains unchanged when time becomes large

enough. This stationary solution for large enough time will be an equilibrium solu-

tion to the IBVP and hence the solution to the given BVP. All of these have been

25 discussed in more detail in section 6.5.

26 Chapter 3

Characterization of Sobolev Spaces and Fractional Powers of Operators

This chapter will be devoted to characterize Sobolev spaces we are considering.

These characterizations are based on the spectral theory of unbounded self-adjoint operators on a Hilbert space having compact resolvents. As all Sobolev spaces we consider are Hilbert spaces, the first of these characterizations will be a more general characterization of the simplest Sobolev space L2(Ω). Then we will introduce new expressions for operator A on L2(Ω) and fractional powers of A.

3.1 Characterization of L2(Ω)

The linear unbounded self-adjoint uniformly elliptic operator A on L2(Ω) has a compact resolvent because (A−λI)−1 is compact in L2(Ω) for every complex λ∈ / σ(A) due to inverse operator theorem (Theorem 8.2, [17]). Thus Theorem 1.8.3 in [17]

∞ 2 implies that there is a complete orthonormal basis {wj}j=1 for L (Ω) of eigenfunctions

∞ of A corresponding to the eigenvalues {λj}j=1 with |λj| → ∞ as j → ∞. Moreover, each wj is smooth on Ω and σ(A) = σp(A), the point spectrum of A. Since A is symmetric and positive definite, all eigenvalues of A are real, positive

27 and 0 < λ1 ≤ λ2 ≤ ..... ≤ λj ≤ .... (counting multiplicity) with λj → ∞ as j → ∞. So we can write u ∈ L2(Ω) as a Fourier series

∞ X u = (u, wj)L2 wj. j=1

For simplification of notation, we prefer to write (u, wj)L2 =u ˆj. Then Parseval’s identity, ∞ X (u, v)L2 = uˆjvˆj j=1

yields L2(Ω)-inner product

∞ 1/2 X 2 1/2 kukL2 = (u, u)L2 = { |uˆj| } . j=1

Thus we can characterize L2(Ω) as

∞ ∞ 2 0 X X 2 L (Ω) = {u ∈ D (Ω): u = uˆjwj, |uˆj| < ∞} j=1 j=1

with the above inner product and norm.

3.2 Characterizations of Hα(Ω), Hαβ(Ω) and Frac-

tional Powers of Operators on L2(Ω)

The operator A has the discrete spectrum σ(A) = {λj : j ∈ N} and EA({λj}) =

(u, wj)L2 =u ˆj. So the spectral integral (2.6.1) enables us to define the operator A on L2(Ω) as

∞ ∞ 2 X 2 2 X D(A) = {u ∈ L (Ω): λj |uˆj| < ∞}, Au = λjuˆjwj j=1 j=1

28 and the operator I + A on L2(Ω) as

∞ ∞ 2 X 2 2 X D(I + A) = {u ∈ L (Ω): (1 + λj) |uˆj| < ∞}, (I + A)u = (1 + λj)ˆujwj. j=1 j=1

Furthermore, (2.6.2) enables us to define Aα on L2(Ω) as

∞ ∞ α 2 X 2α 2 α X α D(A ) = {u ∈ L (Ω): λj |uˆj| < ∞},A u = λj uˆjwj j=1 j=1 and I + Aα on L2(Ω) as

∞ ∞ α 2 X α 2 2 α X α D(I + A ) = {u ∈ L (Ω): (1 + λj ) |uˆj| < ∞}, (I + A )u = (1 + λj )ˆujwj. j=1 j=1

α 2 Since I + A is a self-adjoint positive operator on L (Ω) with eigenfunctions wjs

α α µ and corresponding eigenvalues (1+λj )s, and the map x 7→ (1+x ) , x ∈ [0, ∞), µ >

α µ 0 is Borel measurable EA-finite a. e. ( EA({x ∈ [0, ∞) : (1 + x ) = ∞}) = 0), we can use the spectral integral 2.6.1 to define (I + Aα)µ on L2(Ω) by

∞ ∞ α µ 2 X α 2µ 2 α µ X α µ D((I+A ) ) = {u ∈ L (Ω): (1+λj ) |uˆj| < ∞}, (I+A ) u = (1+λj ) uˆjwj j=1 j=1

We can easily notice that D(A) and D(I + A) are equal as subspaces of L2(Ω), but they are not equal as Hilbert spaces with the inner products inducing the graph

P∞ 2 2 P∞ norms. Indeed, D(I +A) ⊂ D(A) as Hilbert spaces because j=1 λj |uˆj| ≤ j=1(1+

2 2 α α λj) |uˆj| . Analogous situations occur for D(A ) and D(I + A ). As an implication

α 2 α/2 of interpolation method we have H (Ω) = [L (Ω),D(I + A)]α/2 = D((I + A) ).

P∞ α 2 P∞ α 2 On the other hand, j=1(1 + λj) |uˆj| < ∞ ⇔ j=1(1 + λj )|uˆj| < ∞. This implies that D((I+Aα)1/2) = D((I+A)α/2) with the corresponding graph norms. This proves Lemma 2.2.1, that is, as Hilbert spaces D((I + A)α/2) and D((I + Aα)1/2) with the inner products that induce the respective graph norms are equal. Thus Hα(Ω) =

29 D((I + Aα)1/2) justifying the definition of Hα(Ω) given in section 2.2.2 and ensuring

the following characterization

∞ α 2 α 1/2 X α 1/2 2 H (Ω) = {u ∈ L (Ω):(I + A ) u = (1 + λj ) uˆjwj ∈ L (Ω)} j=1

with inner product and norm

∞ ∞ X α X α 2 (u, v)α = (1 + λj )ˆujvˆj, kukα = (1 + λj )|uˆj| . j=1 j=1

Thus the interpolation method ensures once again that Hαβ(Ω) = D((I + Aα)β/2).

That is,

∞ αβ 2 α β/2 X α β/2 2 H (Ω) = {u ∈ L (Ω):(I + A ) u = (1 + λj ) uˆjwj ∈ L (Ω)} j=1

with inner product and norm

∞ ∞ X α β X α β 2 (u, v)αβ = (1 + λj ) uˆjvˆj, kukαβ = (1 + λj ) |uˆj| . j=1 j=1

More interestingly, we can easily extend the definition of Hαβ(Ω) for any β > 0 via

implementation of interpolation method. We also notice that {wj} is an orthogonal

α αβ 2 α basis for each of H (Ω) and H (Ω) but not orthonormal because kwjkα = (1 + λj ) 2 α β and kwjkαβ = (1 + λj ) for all j.

3.3 Fractional Powers of Operators on Hαβ(Ω)

α We will prove in section 4.1 that −A generates a C0-semigroup of contractions on

2 L (Ω). Then perturbation of an infinitesimal generator of C0-semigroup by a bounded linear operator (Therem 1.1 of Chapter 3 in [16]) implies that (−I) + (−Aα) =

30 α 2 −(I + A ) also generates a C0-semigroup on L (Ω). But this is not what we want.

α Instead, we want to construct a C0-semigroup of contractions generated by −(I +A )

αβ on H (Ω). The reason behind constructing this semigroup is to contruct a C0- semigroup generated by L on Hαβ(Ω) with the help of a formula 2.5.3 that expresses

−(I+Aα) the C0-semigroup generated by L in terms of e . The problem is it is somehow

α αβ challenging to construct a C0-semigroup generated by −A directly on H (Ω) so that one can not use perturbation of −Aα by the bounded linear operator −I to obtain

α αβ a C0-semigroup generated by −(I + A ) on H (Ω). To overcome this challenge, we use the spectral integral (2.6.1) to define pseudo-differential operators (I + Aα):

D(I + Aα) ⊂ Hαβ(Ω) → Hαβ(Ω) of order 2α by

∞ α αβ α 2 X α β+2 2 D(I + A ) = {u ∈ H (Ω): k(I + A )ukαβ = (1 + λj ) |uˆj| < ∞}, j=1

∞ α X α (I + A )u = (1 + λj )ˆujwj j=1

and (I + Aα)µ : D(I + Aα)µ ⊂ Hαβ(Ω) → Hαβ(Ω), µ > 0, of order 2αµ by

∞ α µ αβ α µ 2 X α β+2µ 2 D(I + A ) = {u ∈ H (Ω): k(I + A ) ukαβ = (1 + λj ) |uˆj| < ∞}, j=1

∞ α µ X α µ (I + A ) u = (1 + λj ) uˆjwj. j=1

31 Chapter 4

Constructions of Semigroups,

Comparison, Regularity &

Boundedness

To prove the existence and uniqueness results stated in section 2.7, we first require to study the existence and uniqueness of solution to the following homogeneous ACPs

 du α  = −A u if t > 0 dt , (4.0.1)  u[0] = u0

 du  = Lu if t > 0 dt (4.0.2)  u[0] = u0

In order to find unique solutions in suitable Sobolev spaces we need to construct, as

α we discussed in section 2.5, C0-semigroups generated by −A and L. Then we also investigate some of the properties of solutions such as the comparison principle and the regularity property.

32 4.1 Constructions of Semigroups and Comparison

We prove in this section that various fractional powers of operators involving A, which we discussed in chapter 2, are infinitesimal generators of C0-semigroups on L2(Ω).

2 Proposition 4.1.1. −A generates a C0 semigroup of contractions on L (Ω).

Proof. Define for each t ≥ 0

∞ −tA X −tλj 2 e u := e uˆjwj, u ∈ L (Ω), j=1

−tA whereu ˆj and wj are as described in section 3.1. This implies that e defines a map

2 2 −tA from L (Ω) to L (Ω) and ||e u||L2 ≤ ||u||L2 for all t ≥ 0. 2 2 −tA P∞ −2tλj 2 Then for any real number a and u, v ∈ L (Ω), we have e u L2 = j=1 e |uˆj| ≤ P∞ j 2 2 j=1 |u | = kukL2 and

∞ −tA X −tλj e (au + v) = e (au\+ v)jwj j=1 ∞ X −tλj = e (auˆj +v ˆj)wj j=1 ∞ ∞ X −tλj X −tλj =a e uˆjwj + e vˆjwj j=1 j=1

=ae−tAu + e−tAv.

Thus e−tA, t ≥ 0 is a one parameter family of bounded linear operators on L2(Ω)

−tA with e L(L2(Ω)) ≤ 1 ∀ t ≥ 0. Furthermore, it satisfies the following properties.

33 2 −tA For every u ∈ L (Ω), it is clear that e u|t=0 = u, and

∞ −tA −sA −tA X −sλj (e ◦ e )u =e ( e uˆjwj) j=1 ∞ X −sλj −tA = e uˆje (wj) j=1 ∞ X −sλj −tλj = e uje wj j=1 ∞ X −(t+s)λj = e uˆjwj j=1

=e−(t+s)Au, t, s ≥ 0.

Given any  > 0, choose N ∈ N such that

∞ X 2 2 |uˆj| <  /2 j=N+1

P∞ 2 (such an N exists becuase j=1 |uˆj| < ∞). With this N we have

∞ 2 2 X e−tAu − u = (e−tλj − 1)ˆu w L2 j j j=1 L2 ∞ X −tλj 2 2 = (e − 1) |uˆj| j=1 N ∞ X −tλj 2 2 X −tλj 2 2 = (e − 1) |uˆj| + (e − 1) |uˆj| j=1 j=N+1 N ∞ −tλN 2 X 2 X 2 ≤(1 − e ) |uˆj| + |uˆj| j=1 j=N+1 since 0 < 1 − e−tλj ≤ 1 − e−tλN for 1 ≤ j ≤ N. We can choose δ(N) > 0 small enough such that we can make the first term on the right side of the above inequality less than

2 −tA 2 2 −tA  /2 for 0 < t < δ(N) and hence e u − u L2 <  , proving that limt↓0 e u = u

34 in L2(Ω).

−tA At this point, we have proved that e , t ≥ 0 is a C0-semigruop of contractions on L2(Ω).

Finally it remains to show that −A is an infinitesimal generator of e−tA, t ≥ 0.

For u ∈ D(A),

2 ∞ 2 e−tAu − u e−tλj − 1 X − (−Au) = ( + λj)ˆujwj t L2 t j=1 L2 ∞ X e−tλj − 1 = ( + λ )2|uˆ |2 t j j j=1 ∞ X 1 tλj = t2λ4( − + ... )|uˆ |2 j 2! 3! j j=1

→ 0 as t ↓ 0

e−tAu − u and so lim exists in L2(Ω) and equals to −Au. Hence −A is an infinites- t↓0 t imal generator of e−tA, t ≥ 0.

This completes the proof.

Proposition 4.1.2. The semigroup e−tA, t ≥ 0 satisfies the comparison principle on

a finite interval [0,T ].

Proof. Since A is uniformly elliptic, by Theorem 7, Chapter 7 in [8] the following

parabolic IBVP

∂tu(t, x) = −Au(t, x) for t > 0 and u(0, x) = u0(x), x ∈ Ω

subject to periodic or Dirichlet boundary condition has a unique smooth solution

u(t, x). The ACP corresponding to the above IBVP is

du = −Au, u[0] = u dt 0 35 By theorem 4.1.1 and semigroup property SG3 in section 2.5, we obtain that this ACP

−tA has a unique solution e u0. By regularity theory, this solution can be as smooth as we want and thus, by Sobolev embedding theorem, it can be made smooth. Therefore,

−tA u(t, x) = e u0(x). For a moment, let us write u(t, x) as u(t, x). Weak maximum principle (see The-

orem 7, Chapter 7 in [8]) for a parabolic PDE implies that the solution u(t, x) on the interval [0,T ) obtains its maximum value on the parabolic boundary ΓT = ΩT − ΩT

(cylinder without top) where ΩT = Ω × (0,T ]. Thus if u0 ≥ 0 a.e. in Ω

u ≥ min u = − max u− = 0 for a. e. x ∈ Ω ΩT ΓT

because on the bottom of the cylinder −u|Ω×{t=0} = −u0 ≤ 0 a. e. in Ω and on the

lateral surface of the cylinder −u|∂Ω×[0,T ) = 0.

−tA Since, for each fixed t ∈ [0,T ], e u0(x) is continuous on Ω, we must have

−tA −tA e u0 = u(x, t) ≥ 0 for all x ∈ Ω. Hence the semigroup e , t ≥ 0 satisfies the comparison principle on [0,T ].

Aα is densely defined closed linear self-adjoint operator on L2(Ω). The dense-

ness of D(Aα) on L2(Ω) is ensured by D(A) ⊂ D(Aα) ⊂ L2(Ω) and interpolation

inequality. It is immediate form (2.5.1) and (2.5.2) that Aα is closed and self-adjoint.

α More importantly, −A generates a C0-semigroup of contractions that satisfies the comparison principle.

α 2 Proposition 4.1.3. −A generates a C0 semigroup of contractions on L (Ω) satis- fying the comparison principle on a finite time interval.

Proof. The proof follows from Theorem 2.5.9. Indeed, we can replace B by −A

and H by L2(Ω) in equations (2.5.1) through (2.5.4) to prove the assertion. As an

alternative similar to the proof of Proposition 4.1.1, we can also proceed by defining

α α −tA P∞ −tλj 2 e u = j=1 e uˆjwj for t ≥ 0 and u ∈ L (Ω). 36 −tAα α Suppose e , t ≥ 0 is a C0-semigroup of contractions generated by −A . We prove that this semigroup satisfies the comparison principle.

2 −tA Since u0 ∈ L (Ω) and u0 ≥ 0 a. e. in Ω, we have, by Proposition 4.1.2, e u0 ≥ 0 for all t ∈ [0,T ]. Then (2.5.3) yields

Z ∞ −tAα −sA e u0 = e ft,α(s)u0 ds ≥ 0 0

−tAα since ft,α(s) given by (2.5.4) is nonnegative for all s ≥ 0. Hence e , t ≥ 0 satisfies the comparison principle on the interval 0 ≤ t ≤ T .

4.2 Regularity and Boundedness of e−tAα

The last theorem together with the observation we made following the semigroup properties in section 2.5 now ensures that the homogeneous ACP (4.0.1) introduced

−tAα in the beginning of chapter 3 has a unique solution u[t] = e u0, t ≥ 0 for u0 ∈

α α α D(A ) and moreover u[t] ∈ D(A ) ∀ t ≥ 0. But kA u[t]kL2 < ∞ ⇒ ku[t]k2α =

α α 2α k(I + A )u[t]kL2 ≤ ku[t]kL2 + kA u[t]kL2 < ∞, so u[t] lies in H (Ω) and hence u[t] lies in Hs(Ω) for 0 ≤ s ≤ 2α. Actually we can improve the regularity of u[t] even

better.

Proposition 4.2.1. If u ∈ L2(Ω) then e−tAα u ∈ Hs(Ω) ∀ s > 0. For instance, we

have the following estimates:

n −tAα  n  t e 2 2nα ≤ √ e (n ∈ N) (4.2.1) L(L (Ω),H ) 2t

and for each µ ∈ (0, 1), there exists a constant Cµ,T such that

µ −tAα µ t e 2 2µα ≤ Cµ,T e (4.2.2) L(L (Ω),H ) t

37 Proof. By [3], we have the following estimates:

If B is a self adjoint and m-dissipative operator on a Hilbert space H, and if x ∈ H

then etBx ∈ D(Bn) for all n = 1, 2, ... and for any t ∈ (0,T ]

n n tB n tB  n  (−B) e x = (B) e x ≤ √ kxk (4.2.3) H H 2t H

and for any µ ∈ (0, 1), there exists Cµ,T such that for any t ∈ (0,T ]

µ µ tB µ (−B) e x ≤ Cµ,T kxk (4.2.4) H t H

α 2 Since −A is self-adjoint and generates a C0 semigroup on L (Ω), its perturbation

α −(I + A ) by the bounded linear operator −I is also self-adjoint and generates a C0 semigroup on L2(Ω). By Lumer-Phillips Theorem 2.5.5, −Aα is m-dissipative. By

(4.2.3), we have n α n −t(I+Aα)  n  (I + A ) e u 2 ≤ √ kuk 2 (4.2.5) L 2t L

and by (4.2.4), for any µ ∈ (0, 1), there exists Cµ,T such that for any t ∈ (0,T ], we have µ α µ −t(I+Aα) µ (I + A ) e u 2 ≤ Cµ,T kuk 2 (4.2.6) L t L

tI P∞ 1 k P∞ 1 k t We know that e = k=1 k! (tI) = { k=1 k! t }I = e I since I is bounded linear operator on L2(Ω). Therefore, using (4.2.5), we get

−tAα α n −tAα e u 2nα = (I + A ) e u L2

α n tI −t(I+Aα) = (I + A ) e e u L2

t α n −t(I+Aα) =e (I + A ) e u L2 n t n  ≤e √ kuk 2 2t L

38 which proves (4.2.1) and using (4.2.6), we get

−tAα α µ −tAα e u 2µα = (I + A ) e u L2

α µ tI −t(I+Aα) = (I + A ) e e u L2

t α µ −t(I+Aα) =e (I + A ) e u L2 µµ ≤C et kuk µ,T t L2 which proves (4.2.2).

We notice that the estimate (4.2.2) shows that e−tAα u ∈ Hs(Ω) for 0 < s < 2α

s −tAα s because we can take µ = 2α in (0, 1). To prove e u ∈ H (Ω) ∀ s ≥ 0, it is thus sufficient to prove that e−tAα u ∈ Hs(Ω) for all s ≥ 2α.

If s ≥ 2α, we can write s = 2mα − r for some positive integer m and real number

r with 0 ≤ r < 2α. The inequality (4.2.1) implies that e−tAα u ∈ H2mα(Ω). But

H2mα(Ω) ⊂ Hs(Ω) since 2mα ≥ s, we must have e−tAα u ∈ Hs(Ω).

We see from Proposition 4.2.1 that if u ∈ L∞(Ω) ⊂ L2(Ω) then e−tAα u ∈

Hs(Ω) ∀ s > 0. For a large enough s > 0, e−tAα u ∈ L∞(Ω). Sobolev Embedding

Theorem then provides us Hs(Ω) ⊂ L∞(Ω). So e−tAα u ∈ L∞(Ω) for u ∈ L∞(Ω).

Thus e−tAα : L∞(Ω) → L∞(Ω) is a bounded linear operator for large enough

t > 0. In the next proposition, we establish an important L∞(Ω)-estimate for such

an operator e−tAα .

∞ −tAα Proposition 4.2.2. If u ∈ L (Ω) then for each t ≥ 0, e u ∞ ≤ kuk∞.

Proof. Define a constant functionu ¯(x) = kuk∞ for x ∈ Ω. Then u ≤ u¯ a. e. Then the integral formula (2.5.1) for Aα yields

sin(απ) Z ∞ Aαu¯ = sα−1(sI + Aα)−1Au¯ ds = 0 π 0

39 Thus (I + Aα)¯u =u ¯. We discussed in the last proposition that −(I + Aα) generates

−t(I+Aα) 2 α a C0-semigroup e , t ≥ 0 on L (Ω). Sinceu ¯ ∈ D(I + A ), by Property SG3 in section 2.5, e−t(I+Aα)u¯ ∈ D(I + Aα) and

−t(I+Aα) d(e u¯) α α α = −(I + Aα)e−t(I+A )u¯ = −e−t(I+A )(I + Aα)¯u = −e−t(I+A )u¯ dt

α du which shows that u[t] = e−t(I+A )u¯ is a solution to the ACP = −u, u[0] =u ¯. dt du On the other hand, the ACP = −u, u[0] =u ¯ has a unique solution u[t] = e−tI u¯. dt By uniqueness of solution to an ACP, we have e−t(I+Aα)u¯ = e−tu¯. By the comparison principle for e−t(I+Aα), we have

e−t(I+Aα)u ≤ e−tu.¯

Since −u = k−uk∞ = kuk∞, replacing u by −u in the last inequality, we get

−e−t(I+Aα)u ≤ e−tu.¯

Thus from the last two inequalities, we can conclude that

−t(I+Aα) −t −t |e u| ≤ e u¯ ≤ e kuk∞ which then yields

−tAα e u ∞ ≤ kuk∞ , hence proving the assertion.

40 4.3 Constructions of Semigroups and Comparison,

revisited

As a next stage of construction of semigroups, we will focus on the construction of

2αβ a C0-semigroup on H (Ω) generated by L and will prove the comparison principle regarding this semigroup. First we start with constructing a C0-semigroup generated by −(I + Aα) on more general Sobolev space Hαs(Ω).

α Proposition 4.3.1. −(I + A ) generates a C0-semigroup of contractions, namely e−t(I+Aα), t ≥ 0, on Hαs(Ω), s > 0 that satisfies the comparison principle.

Proof. Define

∞ −t(I+Aα) X −t(1+λα) αs e u := e j uˆjwj, t ≥ 0, u ∈ H (Ω). j=1

The arguments similar to those we used in the proof of theorem 4.1.1 will enable

−t(I+Aα) αs us to prove that e , t ≥ 0 is a C0-semigroup of contractions on H (Ω) with infinitesimal generator −(I+Aα) (using the Hαs(Ω)-norm instead of the L2(Ω)-norm).

−t(1+Aα) αs Setting Tα(t)u = e u and adapting the formula for the norm on H (Ω) defined in section 3.2, we get

∞ 2 2 X −t(1+λα) kT (t)uk = {e j uˆ }w α αs j j j=1 αs ∞ α X α s −2t(1+λj ) 2 = (1 + λj ) e |uˆj| j=1 ∞ X α s 2 ≤ (1 + λj ) |uˆj| j=1

2 αs = kukαs ∀ t ≥ 0, u ∈ H (Ω).

αs P∞ If u, v ∈ H (Ω) and a is a real number, then au + v = j=1(auˆj +v ˆj)wj and 41 hence

∞ X −t(1+λα) Tα(t)(au + v) = e j (auˆj +v ˆj)wj j=1 ∞ ∞ X −t(1+λα) X −t(γ+λα) =a e j uˆjwj + e j vˆjwj j=1 j=1

=aTα(t)u + Tα(t)v

αs Thus Tα(t) ∈ L(H (Ω)) and kTα(t)kL(Hαs(Ω)) ≤ 1.

αs We also have Tα(0)u = u. For any t1, t2 ≥ 0 and u ∈ H (Ω), we can apply the spectral integral (2.6.2) with f(x) = e−t(1+xα) to get

∞ α X −t2(1+λ ) Tα(t1) ◦ Tα(t2)u =Tα(t1){ e j uˆjwj} j=1 ∞ α X −t2(1+λ ) = e j uˆjTα(t1)wj j=1 ∞ α α X −t2(1+λ ) −t1(1+λ ) = e j uˆje j wj j=1 ∞ α X −(t1+t2)(1+λ ) = e j uˆjwj j=1

=Tα(t1 + t2)u.

αs 2 P∞ α s 2 Let u ∈ H (Ω). Then kukαs = j=1(1 + λj ) |uˆj| < ∞. Also for each fixed −t(1+λα) −t(1+λα) t ≥ 0, 0 < e j < 1 for all j so that {e j − 1}2 < 1 for all j. For given  > 0 choose an integer N such that

∞ X  (1 + λα)s|uˆ |2 < j j 2 j=N+1

42 For such an N, we have

∞ 2 2 X −t(1+λα) kT (t)u − uk = {e j − 1}uˆ w α αs j j j=1 αs ∞ α X α s −t(1+λj ) 2 2 = (1 + λj ) {e − 1} |uˆj| j=1 N α X α s −t(1+λj ) 2 2 = (1 + λj ) {e − 1} |uˆj| j=1 ∞ α X α s −t(1+λj ) 2 2 + (1 + λj ) {e − 1} |uˆj| j=N+1 N ∞ α X X −t(1+λN ) 2 α s 2 α s 2 ≤{1 − e } (1 + λj ) |uˆj| + (1 + λj ) |uˆj| j=1 j=N+1 N −t(1+λα ) 2 X α s 2  <{1 − e N } (1 + λ ) |uˆ | + . j j 2 j=1

The first finite sum on the right of above inequality can be made as small as want,

−t(1+λα ) say less than /2, by choosing t small enough since e N → 1 as t ↓ 0. Thus the right side of above inequality can be made less than . By arbitrariness of  we have

αs Tα(t)u → u as t ↓ 0 in H (Ω). For u ∈ D(I + Aα),

2 2 ∞ −t(1+λα) T (t)u − u e j − 1 α α X α − {−(I + A )u} = {( + (1 + λj )}uˆjwj t αs t j=1 αs 2 ∞ α X 1 t(1 + λj ) = t2(1 + λα)4{ − + ... }2uˆ w j 2! 3! j j j=1 αs ∞ α X 1 t(1 + λj ) =t2 (1 + λα)s+4{ − + ... }2|uˆ |2 j 2! 3! j j=1

→ 0 as t ↓ 0.

T (t)u − u This shows that lim α exists in Hαβ(Ω) and equals to −(I + Aα)u. t↓0 t

43 Hence we have established that Tα(t), t ≥ 0 is a C0-semigroup of contractions on Hαs(Ω) having infinitesimal generator −(I + Aα). Next we will prove that this semigroup satisfies the comparison principle.

Let u ∈ Hαs(Ω)(⊂ L2(Ω)) such that u ≥ 0 a. e. on Ω. Then u(x) = P∞ j=1 uˆjwj(x) ≥ 0 a. e. on x ∈ Ω. We need to prove that

∞ X −t(1+λα) e j uˆjwj ≥ 0 j=1 on Ω, which is equivalent to ∞ X −tλα e j uˆjwj ≥ 0 j=1

But this follows from theorem 4.1.3 since the semigroup e−tAα , t ≥ 0 on L2(Ω) given

α α −tA P∞ −tλj by e u = j=1 e uˆjwj satisfies the comparison principle, that is,

∞ X −tλα e j uˆjwj ≥ 0. j=1

Hence the proof is complete.

As a consequence of previous theorem we will prove the following important result.

Corollary 4.3.2. The nonlinear operator N on Hαs(Ω), s > 0 satisfies the compar- ison principle.

Proof. Since (I + Aα) is a densely defined closed linear sectorial operator on Hαs(Ω), the proof follows from the following integral formula (see section 2.2.6 in [16] or section

1.3.3 in [26])

∞ 1 Z α (I + Aα)−βu = tβ−1e−t(I+A ) u dt, u ∈ Hαs(Ω) (4.3.1) Γ(β) 0 given in theorem 4.3.1. To be more explicit, if u, v ∈ Hαs(Ω) with u(x) ≥ v(x) a.

44 e. x ∈ Ω then u(x) − f(x, u(x)) ≥ v(x) − f(x, v(x)) a. e. x ∈ Ω since the map y 7→ y − f(x, y) is nondecreasing as y 7→ f(., y) is nonincreasing (i.e. y 7→ F (y) is convex). By theorem 4.3.1, e−t(I+Aα){u − f(x, u)} ≥ e−t(I+Aα){v − f(x, v)}. But then the above integral formula yields (I + Aα)−β{u − f(x, u)} ≥ (I + Aα)−β{v − f(x, v)}, that is, N(u) ≥ N(v).

The gist of this section is the following.

Theorem 4.3.3. The linear operator L generates a C0-semigroup of contractions, namely etLu, t ≥ 0, on Hαs(Ω) that satisfies the comparison principle.

tL Proof. The statement that L generates a C0-semigroup of contractions e u, t ≥ 0 on

α 1−β αs tL P∞ −t(1+λj ) H (Ω) can be proved by either defining e u := j=1 e uˆjwj and adapting ideas that we have been using in the last few theorems or applying the formula (2.5.3) together with the semigroup e−t(I+Aα), t ≥ 0. We would rather prefer the latter one because it provides a straight forward proof of the comparison principle.

Applying the formula (2.5.3), we get

  R ∞ −s(I+Aα) α 1−β  e ft,1−β(s) u ds for t > 0, e−t(I+A ) u = 0  u for t = 0,

where ft,1−β(s) given by (2.5.4) is nonnegative for s > 0. Then theorem 2.5.9 implies

α 1−β tL that L = −(I + A ) generates a C0-semigroup of contractions e , s ≥ 0 on Hαs(Ω). The comparison principle for this semigroup follows from the above formula for e−t(I+Aα)1−β u and the comparison principle for e−t(I+Aα), t ≥ 0 (see theorem

4.3.1).

45 4.4 Regularity and Boundeness Properties of etL

The following technical result about the regularity property of etL is very useful to prove the regularity of the solution to (2.7.2) in the main theorem 2.7.4.

Proposition 4.4.1. If u ∈ Hαβ(Ω), then etLu ∈ Hαs(Ω) for any s ≥ β. For instance,

we have n tL  n  e αβ α(β+2nδ) ≤ √ (n ∈ N) (4.4.1) L(H (Ω),H(Ω) ) 2t

where δ = 1 − β and for each µ ∈ (0, 1), there exists a constant Cµ,T such that

µ tL µ e αβ α(β+2µλ) ≤ Cµ,T (4.4.2) L(H (Ω),H(Ω) ) t

Proof. We will apply (4.2.3) and (4.2.4) with B = L and H = Hαβ(Ω).

αβ Since L is self-adjoint and generates a C0-semigroup on H (Ω), by Lumer-Phillips Theorem 2.5.5, L is m-dissipative and therefore by (4.2.3)

n n L  n  (−L) e u ≤ √ kuk (4.4.3) αβ 2t αβ

and by (4.2.4), for any µ ∈ (0, 1), there exists Cµ,T such that for any t ∈ (0,T ]

µ µ L µ (−L) e u ≤ Cµ,T kuk (4.4.4) αβ t αβ

Since −L = (1 + Aα)δ, using (4.4.3), we get

tL tL e u α(β+2nδ) = e u αβ+2αnδ

α nδ tL = (I + A ) e u αβ

n tL = (−L) e u αβ  n n ≤ √ kuk 2t αβ

46 and in similar fashion, using (4.4.4), we get

µ tL µ tL µ e u = (−L) e u ≤ Cµ,T kuk . α(β+2µδ) αβ t αβ

Thus these two estimates establish (4.4.1) and (4.4.2).

The result that etLu ∈ Hαs ∀ s ≥ β follows from (4.4.1) and (4.4.2).

From Proposition 4.4.1, we see that if u ∈ L∞(Ω) ∩ Hαβ(Ω) then etLu ∈ Hαs(Ω),

∀ s ≥ 0. For a large enough s, etLu ∈ L∞(Ω) since Hαβ(Ω) ⊂ L∞(Ω) due to Sobolev

Embedding Theorem. Therefore, using the method analogous to Proposition 3.6 in

[2], we can establish bounds for etL and N on L∞(Ω).

∞ Proposition 4.4.2. If u ∈ L (Ω), then (a) kN(u)k∞ ≤ kuk∞ + kf(., u)k∞ and (b)

tL −t for each t ≥ 0, e u ∞ ≤ e kuk∞.

Proof. Let us define a constant functionu ¯(x) = kuk∞ for x ∈ Ω. Then u ≤ u¯ a.e. Since, by theorem 4.3.3 and corollary 4.3.2, each of etL and N satisfies the comparison principle, so we have N(u) ≤ N(¯u) and etLu ≤ etLu¯. Now we will see how N and etL behave on constant functions.

For y ∈ R, |f(x, y) − f(x, 0)| < Lip(f)|y|, ∀ x ∈ Ω from which it follows that kf(., u)k∞ ≤ Lip(f) kuk∞ + kf(., 0)k∞ < ∞. Let Λ =u ¯ + kf(., u)k∞. Thenu ¯ − f(., u¯) ≤ Λ. So the integral formula for (I+Aα)−β given by (4.3.1) and the comparison principle for e−t(I+Aα) (theorem 4.3.1) imply that (I + Aα)−β(¯u − f(., u¯)) ≤ (I +

Aα)−β(Λ) and so N(¯u) ≤ (I + Aα)−β(Λ).

As the constant function Λ ∈ D(A) , the integral formula (2.5.1) for Aα yields

sin(απ) Z ∞ AαΛ = sα−1(sI + A)−1AΛ ds = 0, 0 < α < 1. π 0

Therefore, (I + Aα)Λ = Λ. This shows that 1 is an eigenvalue of (1 + Aα) with eigenfunction Λ. But then Proposition 10.3 from [17] gives (I + Aα)−βΛ = 1−βΛ = Λ. 47 Thus N(u) ≤ N(¯u) ≤ Λ

Since −u¯ + f(., u¯) ≤ Λ, the repetition of the above reasoning will establish

−N(u) ≤ Λ.

From the last two inequalities, we get |N(u)| ≤ Λ which then implies that

kN(u)k∞ ≤ Λ. This proves (a). To prove (b), we repeat the argument we used in the proof of part (a) (Proposition

10.3 from [17]) to get

L(¯u) = −(I + Aα)1−β(¯u) = −11−βu¯ = −u¯

tL Sinceu ¯ ∈ D(L), by a property of C0-semigroup in section 2.5 (SG3), e u¯ ∈ D(L) and d(etLu¯) = L(etLu¯) = etLLu¯ = −etLu¯ dt du which shows that u[t] = etLu¯ is a solution to the ACP = −u, u[0] =u ¯. dt du On the other hand, the ACP = −u, u[0] =u ¯ has a unique solution u[t] = e−tu¯. dt Thus, by the uniqueness of solution to an ACP, we must have etLu¯ = e−tu¯. But then, by comparison principle for etL, we have

etLu ≤ e−tu.¯

tL Similarly, as we have −u = k−uk∞ = kuk∞ =u ¯ and e is linear, replacing u by −u in the last inequality, we get

−etL ≤ e−tu.¯

48 Thus we can conclude that

tL −t −t |e u| ≤ e u¯ ≤ e kuk∞ which then yields

tL −t e u ∞ ≤ e kuk∞ , hence proving (b).

49 Chapter 5

The Proofs of Main Results

This chapter is basically devoted to establish the proofs of main theorems that we stated in section 2.7.

We will first prove the main results related to L2-gradient. Then we will prove these results related to Hαβ-gradient.

5.1 The Proofs of Main Theorems Under L2-gradient

We introduce an integral form of (2.4.2)

 du[t]  = −Aαu − f(x, u) if t > 0  dt  u[0] = u0,

¯ where f is continuous and bounded on Ω × R such that fy is continuous and bonded, and the map y 7→ f(., y) is nonincreasing. In the next step, we apply Banach Fixed

Point Theorem to prove that a unique solution in this form always exists.

5.1.1 Some Preliminaries

We write u(t, x) := u[t](x) to see that t 7→ u[t] defines a map from a time interval into a function space.

50 Definition 5.1.1 (Local Mild Solution). A function t 7→ u[t] defined by the integral

form of equation (2.4.2)

Z t −tAα −(t−s)Aα u[t] := e u0 − e f(., u[s]) ds, 0 ≤ t ≤ T (5.1.1) 0

is called a local mild solution of the equation on the interval [0,T ].

Definition 5.1.2 (Local Solution). A map t 7→ u[t] defined from [0,T ] into L2(Ω) is

a (strong) local solution to (2.4.2) if there exists T > 0 such that

(a) u[t] ∈ D(Aα) ∀ t ∈ (0,T ],

(b) u[.] ∈ C([0,T ],D(Aα)),

(c) u[.] ∈ C1((0,T ], L2(Ω)) and du[t] (d) u satisfies (2.4.1) on [0,T ], that is, = −Aα(u[t]) − f(., u[t]) for all dt t ∈ (0,T ] and u[0] = u0.

From the last two definitions, we notice that if a local mild solution u satisfies the

conditions (a)-(c) of definition 5.1.2, then u will automatically satisfy the condition

(d) of it and thus becomes a (strong) local solution to (2.4.1) on [0,T ].

Definition 5.1.3 (Global Solution). If u[.] is a local solution to (2.4.1) on [0,T ] for

all T > 0, then we say u[.] a global solution to (2.4.1).

∞ For any u0 ∈ L (Ω) and T > 0, we define a subspace

∞ WT = {u ∈ C([0,T ], L (Ω)) : u[0] = u0} of C([0,T ], L∞(Ω)) with norm

kuk = max ku[t]k . ∞,T 0≤t≤T ∞

Notice that if 0 < t ≤ T then WT ⊆ Wt and ku[s]k∞,t ≤ ku[s]k∞,T . 51 ∞ The reason to consider the subset WT of the Banach space C([0,T ], L (Ω)) is that we want to find a fixed point of a contraction map defined on WT by applying Banach Fixed Point Theorem. In order to achieve this, we will first need to show that

WT is complete under the norm k.k∞,T , then we will explicitly construct a contraction map on WT in such a way that its fixed point is a mild solution of (2.4.2).

Lemma 5.1.4. WT under the norm k.k∞,T is complete.

∞ Proof. It suffices to prove that WT is a closed subset of C([0,T ], L (Ω)). For, let

∞ {un}n=1 be a sequence in WT with un → u. Then un[0] = u0 ∀ n and kun − ukT,∞ → 0 as n → ∞.

In particular, kun[0] − u[0]k∞ ≤ kun − uk∞,T ∀ n. Taking limit as n → ∞, we

∞ have ku0 − u[0]k∞ = 0. This implies that u[0] = u0 in L sense. Also u, the uniform limit of continuous functions on a compact set, is also continuous. Therefore, u ∈ WT , which proves that WT is closed.

Proposition 5.1.5. The map χ : WT → WT defined by

Z t −tAα −(t−s)Aα χ(u[t]) := e u0 − e f(., u[s]) ds 0

is a contraction map for some positive real number T independent of u0.

Proof. We see that χ(u[0]) = u0 and by Proposition 4.2.2

Z t −tAα −(t−s)Aα kχ(u[t])k∞ ≤ e u0 ∞ + e f(., u[s]) ∞ ds 0 Z t ≤ ku0k∞ + kf(., u[s])k∞ ds 0

≤ ku0k + max kf(., u[s])k ∞ 0≤s≤t ∞

< ∞

which implies that χ defines a map on WT . 52 Since |f(x, y1) − f(x, y2)| ≤ Lip(f)|y1 − y2| ∀ x ∈ Ω, ∀ y1, y2 ∈ R, this yields

kf(., u[s]) − f(., v[s])k∞ ≤ Lip(f) ku[s] − v[s]k∞ ∀ u, v ∈ Wt, 0 ≤ s ≤ t (5.1.2)

and hence with the help of Proposition 4.2.2

Z t −(t−s)Aα kχ(u[t]) − χ(v[t])k∞ ≤ e {f(., u[s]) − f(., v[s])} ∞ ds 0 Z t ≤ kf(., u[s]) − f(., v[s])k∞ ds 0 Z t ≤ Lip(f) ku[s] − v[s]k∞ ds 0

≤ t.Lip(f) ku − vk∞,t

1 Choose T = . Since Lip(f) is independent of u , T is independent of u 2Lip(f) 0 0 and for every t with 0 ≤ t ≤ T

1 kχ(u[t]) − χ(v[t])k ≤ ku − vk ∞ 2 ∞,T

Taking supremum over all t ∈ [0,T ] on the left side, we get

1 kχ(u) − χ(v)k ≤ ku − vk ∞,T 2 ∞,T

and hence χ is a contraction map on WT .

Suppose T is as in the last proposition. At this stage, we can guarantee the existence of unique fixed point uf in WT of the contraction map χ and this fixed

n point is globally attracting. Thus limn→∞ χ = uf and χ(uf ) = uf or equivalently

Z t −tAα −(t−s)Aα uf [t] = e u0 − e f(., uf [s]) ds 0

53 ∞ Here uf is, in fact, a mild solution of (2.4.2) in C([0,T ], L (Ω)).

5.1.2 The Proof of Theorem 2.7.1

Now we use the results proved in the previous subsection. Since T is indepen-

dent of initial data u0, we can follow the same method as above starting a solution at t = T instead of starting at t = 0, we can obtain a unique mild solution in

∞ C([T, 2T ], L (Ω)) of (2.4.2) with the initial data replaced by u[T ] = uf [T ]. Then we can apply concatenation (Gluing Lemma) to obtain a unique mild solution on [0, 2T ] of (2.4.2) with u[0] = u0. Repeating this process indefinitely, we can obtain a unique mild solution on [0, ∞) of (2.4.2) with u[0] = u0. We again denote it by uf . Thus

∞ uf ∈ C([0, ∞), L (Ω)) with uf [0] = u0.

We propose uf as a candidate for the required solution of (2.4.2). For this, we

∞ 2 need to improve the regularity of uf . Since f(., uf ) is in L (Ω) and hence in L (Ω),

R t −(t−s)Aα a property of C0-semigroup (SG4) in section 2.5 implies that 0 e f(., uf [s])ds lies in D(Aα) for each t ≥ 0. The estimate

α α kuk2α = k(1 + A )ukL2 ≤ kukL2 + kA ukL2

R t −(t−s)Aα 2α α then implies that u0 and 0 e f(., uf [s])ds lie in H (Ω). Since u0 ∈ D(A ), −tAα α e u0 ∈ D(A ). Thus we have an estimate

Z t −tAα −(t−s)Aα kuf [t]k2α ≤ e u0 2α + e f(., uf [s])ds , t > 0 0 2α

2α 2α ∞ which yields uf ∈ H (Ω). In order for uf [.] to be in C([0, ∞), H (Ω) ∩ L (Ω)), it

54 suffices to show that it lies in C([0, ∞), H2α(Ω)). But then

t α α Z α −t0A −(t−t0)A −(t−s)A uf [t] − uf [t0] =e {e u0 − u0} − e f(., uf [s])ds t0 t0 Z α α −(t0−s)A −(t−t0)A − e {e f(., uf [s]) − f(., uf [s])}ds 0

for t > t0 ≥ 0 and so

α α −t0A −(t−t0)A kuf [t] − uf [t0]k2α ≤ e L(L2(Ω),H2α(Ω)) e u0 − u0 L2 Z t −(t−s)Aα + e L(L2(Ω),H2α(Ω)) kf(., uf [s])kL2 ds t0 t0 Z α α −(t0−s)A −(t−t0)A + e L(L2(Ω),H2α(Ω)) e f(., uf [s]) − f(., uf [s]) L2 ds. 0

−hAα The definition of C0-semigroup implies that limh↓0 e u0 − u0 L2 = 0. For each −tAα 2 2α t > 0, e ∈ L(L (Ω), H (Ω)) due to proposition 4.2.1. We also have kf(., uf )kL2 ≤ 1/2 |Ω| kf(., uf )k∞ where supt0≤s≤t1 kf(., uf [s])k∞ < ∞. These results enable us to make each term on the right side of the last inquality as small as we want by choos-

+ 2α ing t sufficiently close to t0. Thus the map uf [.] lies in C ([0, ∞), H (Ω)). Also the similar arguments that are used to prove this result ensure that uf [.] lies in

− 2α 2α C ((0, ∞), H (Ω)). Hence uf [.] lies in C([0, ∞), H (Ω)). Until now we have only proved the existence of unique mild solution and improved its regularity continuously on [0, ∞). To complete the proof of theorem 2.7.1, we must

1 2 show that the map t 7→ uf [t] is C from (0, ∞) into L (Ω). In order to show this, we will first show thatu ˙ f [.] exists and is continuous on (0, ∞). Since we expect to have du f = −Aαu − f(., u ), we can propose the first derivative of this map by defining dt f f a map

Z t α −tAα α −(t−s)Aα v[t] := −A e u0 − A e f(., uf [s])ds − f(., uf [t]) for t > 0 0

55 We first want to prove that

1 lim (uf [t + h] − uf [t]) − v[t] = 0 h↓0 h L2

For t > 0 and h > 0

1 1 α α (u [t + h] − u [t]) − v[t] ={ (e−hA − I) + Aα}e−tA u h f f h 0 Z t 1 −hAα α −(t−s)Aα − { (e − I) − A }e f(., uf [s]) ds 0 h Z t+h 1 −(t+h−s)Aα − e f(., uf [s]) ds + f(., uf [t]). h t

−tAα α −tAα α Since e u0 ∈ D(A ) and e u0, t ≥ 0 is generated by −A , we have

1 −hAα −tAα α −tAα lim (e − I)e u0 = −A e u0, h↓0 h

under L2(Ω)-convergence, that is,

1 −hAα α −tAα lim { (e − I) + A }e u0 = 0. h↓0 h L2

−hAα R h d −τAα R h α −τAα α −τAα As we have (e − I) = 0 dτ (e ) dτ = 0 A e dτ and −A e −(t−s)Aα −τAα α −(t−s)Aα −(t−s)Aα α e f(., uf [s]) = −e A e f(., uf [s]) since e f(., uf [s]) ∈ D(A ) (see Proposition 4.2.1), we obtain

Z t 1 −hAα α −(t−s)Aα { (e − I) − A }e f(., uf [s])ds 0 h L2 Z t Z h 1 −τAα α −(t−s)Aα ≤ { e dτ − I}A e f(., uf [s]) ds. 0 h 0 L2

α −(t−s)Aα 2 1 R h −τAα α −(t−s)Aα Since A e f(., uf [s]) ∈ L (Ω) and limh↓0 h 0 e (A e f(., uf [s])) dτ = α −(t−s)Aα 2 A e f(., uf [s]) in L (Ω), the integrand on right side of above inequality ap- proaches 0 as h ↓ 0. The continuity of the ingrand on the left side of the inequality on 56 the interval [0, t] enables us to switch the limit and the integral sign. Thus we have

Z t 1 −hAα α −(t−s)Aα lim { (e − I) − A }e f(., uf [s])ds = 0. h↓0 0 h L2

Substituting s = t + τ, we get

Z t+h 1 −(t+h−s)Aα e f(., uf [s])ds − f(., uf [t]) h t Z h 1 −(h−τ)Aα = e f(., uf [τ + t])dτ − f(., uf [t]) h 0 Z h 1 −(h−τ)Aα = { e f(., uf [t])dτ − f(., uf [t])} h 0 Z h 1 −(h−τ)Aα + e {f(., uf [τ + t]) − f(., uf [t])}dτ. h 0

−(h−τ)Aα Let  > 0 be given. Since e f(., uf [t]) − f(., uf [t]) L2 → 0 as h → 0 for −(h−τ)Aα 0 ≤ τ ≤ h, we can make e f(., uf [t]) − f(., uf [t]) L2 less than  by choosing h sufficiently close to 0. For such an h,

Z h 1 −(h−τ)Aα { e f(., uf [t])dτ − f(., uf [t])} h 0 L2 Z h 1 −(h−τ)Aα = {e f(., uf [t]) − f(., uf [t])}dτ h 0 L2 Z h 1 −(h−τ)Aα ≤ e f(., uf [t]) − f(., uf [t]) L2 dτ h 0 < .

∞ Also the continuity of the map t 7→ uf [t] from the interval [0, ∞) into L (Ω) enables p us to make kuf [τ + t] − uf [t]k∞ < /{ |Ω|Lip(f)} whenever τ ∈ [0, δ], for some

57 sufficiently small δ > 0. Thus for any h ≤ δ,

Z h 1 −(h−τ)Aα e {f(., uf [τ + t]) − f(., uf [t])}dτ h 0 L2 Z h 1 −(h−τ)Aα ≤ e L(L2(Ω)) kf(., uf [τ + t]) − f(., uf [t])kL2 dτ h 0 1 Z h p ≤ |Ω| kf(., uf [τ + t]) − f(., uf [t])k∞ dτ h 0 p 1 Z h ≤ |Ω|Lip(f) kuf [τ + t] − uf [t]k∞ dτ h 0 < .

It follows from the last few estimates that

Z t+h 1 −(t+h−s)Aα lim e f(., uf [s])ds − f(., uf [t]) = 0. h↓0 h t L2

Thus we have proved that

1 lim (uf [t + h] − uf [t]) − v[t] = 0. h↓0 h L2

+ + This shows thatu ˙ f [t] exists andu ˙ f [t] = v[t] on (0, ∞). Furthermore, the continuity of

α −tAα R t α −(t−s)Aα each of the maps t 7→ A e u0, t 7→ 0 A e f(., uf [s])ds and t 7→ f(., uf [t]) 2 + 2 from [0, ∞) into L (Ω) ensures the continuity ofu ˙ f [t] on (0, ∞) undeder L (Ω)- − convergence. Similarly, we can proveu ˙ f [t] = v[t]. Henceu ˙ f [t] = v[t] on (0, ∞) and

1 ∞ 2 uf ∈ C ((0, ∞), L (Ω) ∩ L (Ω)). At this stage, we have completed the proof of theorem 2.7.1.

5.1.3 The Proof of Theorem 2.7.3

Now we prove Theorem 2.7.3, the comparison principle for the equation (2.7.1), on the time interval [0,T ], first using the iteration method and then extending it on

58 [0, ∞) by the help of independence of T on u0.

For any u ∈ WT , we have

Z t χ(u([t]) = e−tAα (u[0]) − e−(t−s)Aα f(., u[s]) ds, t ∈ [0,T ]. 0

−tAα Set u[t] = e u0, t ∈ [0,T ]. By Proposition 4.2.2, u ∈ WT since t 7→ u[t] is

∞ ∞ continuous on [0,T ] under L (Ω)-norm and u[0] = u0 ∈ L (Ω) . Then we obtain iterations of χ at u as follows:

−tAα χ(e u0) =χ(u[t]) Z t =e−tAα u[0] − e−(t−s)Aα f(., u[s]) ds 0 Z t −tAα −(t−s)Aα −sAα =e u0 − e f(., e u0) ds, 0 2 −tAα χ (e u0) =χ(v[t]), where v[t] = χ(u[t]) Z t =e−tAα v[0] − e−(t−s)Aα f(., v[s]) ds 0 Z t −tAα −(t−s)Aα =e u0 − e f(., χ(u[s])) ds 0 Z t −tAα −(t−s)Aα −sAα =e u0 − e f(., χ(e u0)) ds, 0

n −tAα Continuing in this way, as χ (e u0) ∈ WT for each n = 0, 1, 2,... , we have

n+1 −tAα n −tAα χ (e u0) =χ(χ (e u0)) Z t −tAα n −tAα −(t−s)Aα n −sAα =e (χ (e u0)|t=0) − e f(., χ (e u0)) ds 0 Z t −tAα −(t−s)Aα n −sAα =e u0 − e f(., χ (e u0)) ds, 0 for all n = 0, 1, 2,... .

The next proposition shows that each of these iterations satisfies the comparison

59 principle.

∞ Proposition 5.1.6. Let u0, v0 ∈ L (Ω) such that u0 ≥ v0 a.e. in Ω. Then

n −tAα n −tAα χ (e u0) ≥ χ (e v0) ∀ n ∈ N, ∀ t ∈ [0,T ].

Proof. The comparison principle for e−tAα and the nonincreasing property of f at

−tAα −tAα −tAα functional component y imply that e u0 ≥ e v0 ∀ t ∈ [0,T ] and f(e u0) ≤

−tAα f(e v0) ∀ t ∈ [0,T ]. From this, it follows that

−tAα −tAα χ(e u0) ≥ χ(e v0) ∀ t ∈ [0,T ] showing that the result holds true for n = 1.

Let us assume, by induction, that the result holds true for some positive integer n. The comparison and nonincreasing arguments of the previous paragraph yield, for each t ∈ [0,T ],

n+1 −tAα n −tAα χ (e u0) =χ(χ (e u0))

n −tAα ≥χ(χ (e v0))

n+1 −tAα =χ (e v0).

Hence the assertion follows from the induction.

∞ Theorem 5.1.7. Let u0, v0 ∈ L (Ω) such that u0 ≥ v0 a. e. in Ω. Then uf [t] ≥ vf [t] for a. e. in Ω, ∀ t ∈ [0,T ] and hence ∀ t ∈ [0, ∞).

Proof. From the previous propsition 5.1.6, we have

n −tAα n −tAα χ (e u0) ≥ χ (e v0) ∀ n ∈ N, ∀ t ∈ [0,T ].

60 This implies that

n −tAα n −tAα lim χ (e u0) ≥ lim χ (e v0) ∀ t ∈ [0,T ]. n→∞ n→∞

Since uf and vf are the globally attracting fixed points of χ corresponding to the initial data u0 and v0, for each t ∈ [0,T ]

n −tAα n −tAα lim χ (e u0) = uf [t] and lim χ (e v0) = vf [t] in WT . n→∞ n→∞

n −tAα n −tAα But |χ (e u0(x))−uf [t](x)| ≤ χ (e u0) − uf ∞,T ∀ t ∈ [0,T ] for almost every x ∈ Ω,

n −tAα lim χ (e u0(x)) = uf [t](x) ∀ t ∈ [0,T ] for almost every x∈ Ω. n→∞

Similarly,

n −tAα lim χ (e v0(x)) = vf [t](x) ∀ t ∈ [0,T ] for almost every x∈ Ω. n→∞

Therefore uf [t](x) ≥ vf [t](x) ∀ t ∈ [0,T ] for almost every x ∈ Ω.

Now it remains to show that we can extend the comparison of uf and vf on the entire time interval [0, ∞).

Due to Proposition 5.1.5, let [0,T1] with T1 > 0 be an interval of existence corre- sponding to the initial condition u0 for which uf is the globally attracting fixed point

of WT1 and [0,T2] with T2 > 0 be an interval of existence corresponding to the initial

condition v0 for which vf is the globally attracting fixed point of WT2 . By choosing

T = min{T1,T2}, we can prove, as shown in the proof of Theorem 2.7.1, that there exist unique solutions uf and vf of equation (2.4.2) on [0,T ] corresponding to the initial conditions u0 and v0 and hence of (2.7.1) on [0,T ] corresponding to the same

61 initial conditions. Then the first part we have just proved implies that uf [t] ≥ vf [t] for all t ∈ [0,T ] and a.e. in Ω.

Since T is independent of both u0 and v0, we can continue to find, as we have shown in subsection 4.1.2, the unique solutions uf and vf of (2.7.1) on [T, 2T ] corresponding

to u0 and v0, then by repeating the arguments of this subsection until the first part of this theorem, we can prove that uf ≥ vf for all t ∈ [T, 2T ] and for almost every x in

Ω. By concatenating the respective solutions, we can extend the existence of uf and

vf on [0, 2T ] and hence their comparison on [0, 2T ]. As we can keep extending the

existence of uf and vf on the sequence of time intervals [0,T ], [0, 2T ], [0, 3T ],..., we will be able to continue establishing their comparison on this sequence of time

intervals. Thus we can establish the comparison principle for (2.7.1) on the entire

time interval [0, ∞).

5.2 The Proofs of Main Theorems Under Hαβ-gradient

In this section, we are going to extend the analogous results to Hαβ-gradients

what we established for L2-gradients in section 5.1

5.2.1 Some Preliminaries (Revisited)

Let us recall (2.7.2)

 du[t]  = Lu[t] + Nu[t] for t > 0  dt  u[0] = u0

Now we give an integral form of this equation below.

Definition 5.2.1 (Local Mild Solution). A map t 7→ u[t] defined by an integral form

62 of equation (2.4.1) given by

Z t tL (t−s)L u[t] := e u0 + e N(u[s]) ds, 0 ≤ t ≤ T (5.2.1) 0

is called a local mild solution to (2.4.1) on [0,T ].

Definition 5.2.2 (Local Solution). A map t 7→ u[t] defined from [0,T ] into H2αβ(Ω)

is a (strong) local solution to (2.4.1) if there exists T > 0 such that

(a) u[t] ∈ D(L) ∀ t ∈ (0,T ],

(b) u[.] ∈ C([0,T ],D(L)),

(c) u[.] ∈ C1((0,T ], H2αβ(Ω)) and du[t] (d) u satisfies (2.4.1) on [0,T ], that is, = L(u[t]) + N(u[t]) for all t ∈ (0,T ] dt and u[0] = u0.

From the last two definitions, we notice that if a local mild solution u satisfies the

conditions (a)-(c) of definition 5.2.2, then u will automatically satisfy the condition

(d) of it and thus becomes a (strong) local solution to (2.4.1) on [0,T ].

Definition 5.2.3 (Global Solution). If u[.] is a local solution to (2.4.1) on [0,T ] for

all T > 0, then we say u[.] a global solution to (2.4.1).

Proposition 5.2.4. The map Ψ: WT → WT defined by

Z t tL (t−s)L Ψ(u[t]) := e u0 + e N(u[s]) ds, 0 ≤ t ≤ T 0

is a contraction map for some positive real number T independent of u0 and hence has a fixed point that is globally attracting.

63 Proof. We have Ψ(u[0]) = u0. It follows from proposition 4.4.2 that

Z t tL (t−s)L kΨ(u[t])k∞ ≤ e u0 ∞ + e N(u[s]) ∞ ds 0 Z t −t −(t−s) ≤ e ku0k∞ + e kN(u[s])k∞ ds 0 Z t −t −(t−s) ≤ e ku0k∞ + e (ku[s]k∞ + kf(., u[s])k∞) ds 0 −t −t ≤ e ku0k + (kuk + max kf(., u[s])k )(1 − e ) ∞ ∞,T 0≤s≤t ∞

< ∞.

Thus Ψ defines a map on WT . Adapting the argument of the proof of the first part of proposition 4.4.2, we obtain

kN(u[s]) − N(v[s])k∞ ≤ ku[s] − v[s]k∞ + kf(., u[s]) − f(., v[s])k∞ . (5.2.2)

From the estimates (5.2.2) and (5.1.2), we have

Z t (t−s)L kΨ(u[t]) − Ψ(v[t])k∞ ≤ e {N(u[s]) − N(v[s])} ∞ ds 0 Z t (t−s)L ≤ e L(L∞(Ω)) kN(u[s]) − N(v[s])k∞ ds 0 Z t −(t−s) ≤ e (ku[s] − v[s]k∞ + kf(., u[s]) − f(., v[s])k∞) ds 0 ≤ (ku[s] − v[s]k + max kf(., u[s]) − f(., v[s])k )(1 − e−t) ∞,t 0≤s≤t ∞

≤ (ku[s] − v[s]k∞,t + Lip(f) ku[s] − v[s]k∞,t)t

≤ t{1 + Lip(f)} ku[s] − v[s]k∞,t .

1 Choose T = . As we see that Lip(f) is independent of u , T is 2{1 + Lip(f)} 0

64 independent of u0 and for every t with 0 ≤ t ≤ T

1 kΨ(u[t]) − Ψ(v[t])k ≤ ku − vk . ∞ 2 ∞,T

Taking supremum over all t ∈ [0,T ] on the left hand side, we get

1 kΨ(u) − Ψ(v)k ≤ ku − vk . ∞,T 2 ∞,T

This shows that Ψ is a contraction map on WT .

Therefore, by Banach Fixed Point Theorem, Ψ has a unique fixed point uF in WT that is globally attracting.

5.2.2 The Proof of Theorem 2.7.2

The last proposition implies that

Z t tL (t−s)L uF [t] = e u0 + e N(uF [s]) ds, t ∈ [0,T ] 0

is a unique mild solution to (2.4.1) in C([0,T ], L∞(Ω)). Then the similar arguments

we applied in the proof of Theorem (2.4.2) in section 5.1.2 will also confirm that uF [.] is a unique mild solution to (2.4.1) in C([0, ∞), L∞(Ω)).

2 As the L -gradient counterpart, we propose uF as a candidate for the required solution to (2.4.1). For this, we need to prove that uF satisfies the conditions (a)-(c) of Definition 4.2.2.

∞ 2 First we will prove that uF [t] lies in D(L) for t ≥ 0. Since L (Ω) ⊂ L (Ω), we

65 have

α −β kN(uF [.])k2αβ = (I + A ) (uF [.] − f(., uF [.]) 2αβ

α β α −β = (I + A ) (I + A ) (uF [.] − f(., uF [.]) L2

= kuF [.] − f(., uF [.])kL2 p ≤ |Ω|(kuF [.]k∞ + kf(., uF [.])k∞).

2αβ Thus N(uF [t]) ∈ H (Ω) for t ≥ 0. From Proposition 4.4.1, the last estimate, and

2αβ 2αβ the fact that the linear operator L : H (Ω) → H (Ω) generates a C0-semigroup of contractions etL, t ≥ 0 (see Theorem 4.3.3), we have

Z t tL (t−s)L kuF [t]k2αβ ≤ e u0 2αβ + e N(uF [s]) 2αβ ds 0 Z t tL (t−s)L ≤ e u0 2αβ + e L(H2αβ (Ω)) kN(uF [s])k2αβ ds 0 Z t tL p ≤ e u0 2αβ + |Ω|(kuF [s]k∞ + kf(., uF [s])k∞) ds 0 tL p ≤ e u0 2αβ + t |Ω|(kuF k∞,t + kf(., uF )k∞,t).

2αβ This shows that uF [t] ∈ H (Ω) for every t > 0. Since the linear operator L : H2αβ(Ω) → H2αβ(Ω) has a property that for any u ∈ H2αβ(Ω)

α 1−β α (1−β)+β α kLuk2αβ = (I + A ) u 2αβ = (I + A ) u L2 = k(I + A )ukL2 = kuk2α ,

tL 2αβ so u ∈ D(L) if only if kuk2α < ∞. As e , t ≥ 0 is C0-semigroup on H (Ω) and

2αβ N(uF [t]) ∈ H (Ω) for t ≥ 0, by a property of C0-semigroup (SG4, section 1.5), we

66 R t (t−s)L tL must have 0 e N(uF [s]) ds ∈ D(L) for t ≥ 0. Also e u0 ∈ D(L). Thus

Z t tL (t−s)L kuF k2α ≤ e u0 2α + e N(uF [s]) ds 0 2α Z t tL (t−s)L = Le u0 2αβ + L e N(uF [s]) ds 0 2αβ <∞.

2α Thus uF [t] ∈ D(L) and uF [t] ∈ H (Ω) for t ≥ 0 as well.

Now we prove that the map t 7→ uF [t] is a continuous map from [0, ∞) into

2α ∞ 2α H (Ω) ∩ L (Ω). To prove this, it suffices to show that uF [.] ∈ C([0, ∞), H (Ω)).

For t > t0 ≥ 0,

Z t t0L (t−t0)L (t−s)L uF [t] − uF [t0] =e {e u0 − u0} + e N(uF [s]) ds t0 Z t0 (t0−s)L (t−t0)L + e {e N(uF [s]) − N(uF [s])} ds, 0 and so

t0L (t−t0)L kuF [t] − uF [t0]k2α ≤ e L(H2αβ (Ω),H2α(Ω)) e u0 − u0 2αβ Z t (t−s)L + e L(H2αβ (Ω),H2α(Ω)) kN(uF [s])k2αβ ds t0 Z t0 (t0−s)L (t−t0)L + e L(H2αβ (Ω),H2α(Ω)) e N(uF [s]) − N(uF [s]) 2αβ ds. 0

hL The definition of C0-semigroup implies that limh↓0 e u0 − u0 2αβ = 0. For each t > 0, etL ∈ L(H2αβ(Ω), H2α(Ω)) due to proposition 4.2.1. These results enable us to make each term on the right side of the last inequality as small as we want by

+ 2α choosing t sufficiently close to t0. Thus the map uF [.] lies in C ([0, ∞), H (Ω)).

Also the similar arguments that are used to prove this result ensure that uF [.] lies in

− 2α 2α C ((0, ∞), H (Ω)). Hence uf [.] lies in C([0, ∞), H (Ω)).

67 2αβ Finally, we prove that the map t 7→ uF [t] from (0, ∞) into H (Ω) is continuously du differentialble. With the help of = Lu+Nu, we use as an ansatz for the derivative dt of this map as the following

Z t tL (t−s)L v[t] := Le u0 + L e N(uF [s])ds + N(uF [t]) for t > 0. 0

We first need to prove that

1 lim (uF [t + h] − uF [t]) − v[t] = 0. h↓0 h 2αβ

For t > 0 and h > 0,

1 1 (u [t + h] − u [t]) − v[t] ={ (ehL − I) − L}etLu h F F h 0 Z t 1 hL (t−s)L + { (e − I) − L} e N(uF [s]) ds h 0 Z t+h 1 (t+h−s)L + e N(uF [s]) ds − N(uF [t]), h t

1 hL where the exchange of the order of integral sign and the operator h (e −I) to obtain 1 hL the second line is possible due to the fact that the bounded linear operator h (e −I) on H2αβ(Ω) is closed.

Recalling the definition of the infinitesimal generator of a C0-semigroup, we know ehL − I that u ∈ D(L) implies lim u = Lu in H2αβ(Ω). Also by C -semigroup h↓0 h 0 tL R t (t−s)L properties, e u0 and 0 e N(uF [s]) ds belong to D(L). Thus we have

1 hL tL lim { (e − I) − L}e u0 = 0 and h↓0 h 2αβ

Z t 1 hL (t−s)L lim { (e − I) − L} e N(uF [s]) ds = 0. h↓0 h 0 2αβ

68 Substituting s = t + τ, we get

Z t+h 1 (t+h−s)L e N(uF [s]) ds − N(uF [t]) h t Z h 1 (h−τ)L = e N(uF [τ + t]) dτ − N(uF [t]) h 0 Z h 1 (h−τ)L = { e N(uF [t]) dτ − N(uF [t])} h 0 Z h 1 (h−τ)L + e {N(uF [t + τ]) − N(uF [t])}dτ. h 0

We analyze two expressions on the right side of the above equation separately.

(h−τ)L Let  > 0 be given. Since e N(uF [t]) − N(uF [t]) 2αβ → 0 as h → 0 for (h−τ)L 0 ≤ τ ≤ h, we can make e N(uF [t]) − N(uF [t]) 2αβ less than /2 by choosing h sufficiently close to 0. For such an h,

Z h 1 (h−τ)L { e N(uF [t])dτ − N(uF [t])} h 0 2αβ Z h 1 (h−τ)L = {e N(uF [t]) − N(uF [t])}dτ h 0 2αβ Z h 1 (h−τ)L ≤ e N(uF [t]) − N(uF [t]) 2αβ dτ h 0 < /2.

2αβ Also the continuity of the map t 7→ uF [t] from the interval [0, ∞) into H (Ω)  enables us to make kuF [τ + t] − uF [t]k < whenever τ ∈ [0, δ], 2αβ 2p|Ω|(1 + Lip(f))

69 for some sufficiently small δ > 0. Thus for any positive h ≤ δ,

Z h 1 (h−τ)L e {N(uF [τ + t]) − N(uF [t])}dτ h 0 2αβ Z h 1 (h−τ)L ≤ e L(H2αβ (Ω)) kN(uF [τ + t]) − N(uF [t])k2αβ dτ h 0 1 Z h p ≤ |Ω|{kuF [τ + t] − uF [t]k∞ + kf(., uF [τ + t]) − f(., uF [t])k∞} dτ h 0 1 Z h p ≤ |Ω|(1 + Lip(f)) kuF [τ + t] − uF [t]k∞ dτ h 0 p 1 Z h ≤ |Ω|(1 + Lip(f)) kuF [τ + t] − uF [t]k∞ dτ h 0 < /2.

From the results we proved in the last paragraph, it follows that

Z t+h 1 (t+h−s)L lim e N(uF [s])ds − N(uF [t]) = 0. h↓0 h t 2αβ

Thus we have proved that

1 lim (uF [t + h] − uF [t]) − v[t] = 0. h↓0 h 2αβ

+ + This shows thatu ˙ F [t] exists andu ˙ F [t] = v[t] on (0, ∞). Furthermore, the continuity

tL R t (t−s)L of each of the maps t 7→ Le u0, t 7→ L 0 e N(uF [s]) ds, and t 7→ N(uF [t]) from 2αβ + 2αβ [0, ∞) into H (Ω) ensures the continuity ofu ˙ F [t] from (0, ∞) into H (Ω). Hence

+ 1 ∞ 2αβ u˙ F [t] =u ˙ F [t] on (0, ∞) and uF ∈ C ((0, ∞), L (Ω) ∩ H (Ω)). Hence we have completed the proof of theorem 2.7.2.

5.2.3 The Proof of Theorem 2.7.4

In this subsection, we will prove the comparison principle for the equation (2.7.2)

on the time interval [0,T ] using the iteration method. Finally we extend it on [0, ∞)

70 using the method analogous to the one used for proving the comparison principle for

(2.7.1) in subsection 4.2.

We know that for any u ∈ WT

Z t Ψ(u([t]) = etL(u[0]) + e(t−s)LN(u[s]) ds, t ∈ [0,T ]. 0

tL Set u[t] := e u0, t ∈ [0,T ]. Due to Proposition 4.4.2, u ∈ WT . Now we compute iterations of Ψ at u as follows

tL Ψ(e u0) =Ψ(u[t]) Z t =etLu[0] + e(t−s)LN(u[s]) ds 0 Z t tL (t−s)L sL =e u0 + e N(e u0) ds, 0 Z t 2 tL tL (t−s)L Ψ (e u0) =e Ψ(u[0]) + e N(Ψ(u[s])) ds 0 Z t tL (t−s)L sL =e u0 + e N(Ψ(e u0)) ds, 0

n tL Continuing in this way, if Ψ (e u0) ∈ WT for each n = 0, 1, 2,... , then we have

n+1 tL n tL Ψ (e u0) =Ψ(Ψ (e u0)) Z t tL n tL (t−s)L n sL =e (Ψ (e u0)|t=0) + e N(Ψ (e u0)) ds 0 Z t tL (t−s)L n sL =e u0 + e N(Ψ (e u0)) ds, 0 for all n = 0, 1, 2,... .

Now we will show that each of these iterations satisfies the comparison principle.

∞ Proposition 5.2.5. Let u0, v0 ∈ L (Ω) such that u0 ≥ v0 a.e in Ω. Then

n tL n tL Ψ (e u0) ≥ Ψ (e v0) ∀ n ∈ N, ∀ t ∈ [0,T ].

tL tL tL Proof. The comparison principle for e and N imply that e u0 ≥ e v0 ∀ t ∈ [0,T ] 71 tL tL and N(e u0) ≥ N(e v0) ∀ t ∈ [0,T ]. From this it follows that

tL tL Ψ(e u0) ≥ Ψ(e v0) ∀ t ∈ [0,T ]

showing that the result holds true for n = 1.

Let us assume, by induction, that the result holds true for some positive integer

n. The result proved in the last paragraph for n = 1 together with the induction

hypothesis, we have for each t ∈ [0,T ],

n+1 tL n tL Ψ (e u0) =Ψ(Ψ (e u0))

n tL ≥Ψ(Ψ (e v0))

n+1 tL =Ψ (e v0).

Hence the assertion follows from the induction.

∞ Theorem 5.2.6. Let u0, v0 ∈ L (Ω) such that u0 ≥ v0 a. e. in Ω. Then uF [t] ≥

vF [t] for a. e. in Ω, ∀ t ∈ [0,T ] and hence ∀ t ∈ [0, ∞).

Proof. From Proposition 5.2.5, we have

n tL n tL Ψ (e u0) ≥ Ψ (e v0) ∀ n ∈ N, ∀ t ∈ [0,T ].

This implies that

n tL n tL lim Ψ (e u0) ≥ lim Ψ (e v0) ∀ t ∈ [0,T ]. n→∞ n→∞

Since uF and vF are the globally attracting fixed points of Ψ corresponding to the initial data u0 and v0, for each t ∈ [0,T ]

n tL n tL lim Ψ (e u0) = uF [t] and lim Ψ (e v0) = vF [t] in WT . n→∞ n→∞ 72 n tL n tL But |Ψ (e u0(x)) − uF [t](x)| ≤ Ψ (e u0) − uF ∞,T ∀ t ∈ [0,T ] for almost every x ∈ Ω and thus

n tL lim Ψ (e u0(x)) = uF [t](x) ∀ t ∈ [0,T ] for almost every x ∈ Ω. n→∞

Similarly,

n tL lim Ψ (e v0(x)) = vF [t](x) ∀ t ∈ [0,T ] for almost every x ∈ Ω. n→∞

Therefore uF [t](x) ≥ vF [t](x) ∀ t ∈ [0,T ] for almost every x ∈ Ω.

Now it remains to show that we can extend the comparison of uF and vF on the entire time interval [0, ∞).

Due to proposition 5.2.4, let [0,T1] with T1 > 0 be an interval of existence corre- sponding to the initial condition u0 for which uF is the globally attracting fixed point

of Ψ in WT1 and [0,T2] with T2 > 0 be an interval of existence corresponding to the

initial condition v0 for which vF is the globally attracting fixed point of Ψ in WT2 . By choosing T = min{T1,T2}, we can prove, as shown in the proof of Theorem 2.7.2 in subsection 5.2.2, that there exist unique solutions uF and vF of equation (2.4.1) on

[0,T ] corresponding to the initial conditions u0 and v0 and hence of (2.7.1) on [0,T ] corresponding to the same initial conditions. Then the first part we have just proved gives uF ≥ vF for all t ∈ [0,T ] and for almost every x in Ω.

Since T is independent of both u0 and v0, we can continue to find, as we have shown in subsection 5.2.2, the unique solutions uF and vF of (2.7.2) on [T, 2T ] corresponding to u0 and v0, then by repeating the arguments of this subsection until the first part of this theorem, we can prove that uF ≥ vF for all t ∈ [T, 2T ] and for almost every x in

Ω. By concatenating the respective solutions, we can extend the existence of uF and vF on [0, 2T ] and hence their comparison on [0, 2T ]. As we can keep extending the

73 existence of uF and vF on the sequence of time intervals [0,T ], [0, 2T ], [0, 3T ],..., we will be able to continue establishing their comparison on this sequence of time intervals. Thus we can establish the comparison principle for (2.7.1) on the entire time interval [0, ∞).

74 Chapter 6

Applications to Semilinear Pseudo

Differential Equations

In this chapter, we are going to apply the existence, uniqueness, and comparison results we established throughout Chapters 2-4 to solve nonlinear partial and pseudo differential equations subject to periodic boundary conditions. We will use these results to study a special class of solutions to the Euler-Lagrange equations for the variational problem Z 1 α/2 2 Eα(u) = |A u| + F (x, u)dx. (6.0.1) Rd 2

Here F (x, u) is as introduced in 2.1 with additional assumption that f is Zd+1−periodic and the coefficient functions aij of the elliptic operator A (see (2.1.2) for the defini- tion) are Zd−periodic. With these assumptions, the variational problem (6.0.1) can

d+1 be reduced to the one on a torus. Indeed, we writex ¯ ∈ R asx ¯ = (x, xd+1), where

d x = (x1, x2, . . . , xd) ∈ R , and assume that

¯ ¯ d+1 f(¯x + k) = f(¯x) ∀ k = (k, kd+1) ∈ Z ,

ij ij d d a (x + k) = a (x) ∀ k ∈ Z , ∀ x ∈ R .

75 6.1 Nonself-intersecting Solutions

We require that our variational problem is invariant under the translationx ¯ 7→

¯ ¯ d+1 x¯+k where k ∈ Z . This means that any solution xd+1 = u(x) to the corresponding Euler-Lagrange equation (2.1.1), that is

Aαu + f(., u) = 0 (6.1.1)

(the equation satisfied by the critical points of Eα), can be transformed into another ¯ d+1 solution xd+1 = u(x + k) − kd+1 ∀ k ∈ Z to the variational problem, giving the same hypersurface on the torus Td+1.

Definition 6.1.1. A function u : Rd → R is said to have nonself-intersecting (or

d+1 Birkhoff) property on T if the hypersurface xd+1 = u(x) and its translation under the Zd+1-action given by

d ¯ d+1 Tk¯u(x) = u(x + k) − kd+1 ∀ x ∈ R , k = (k, kd+1) ∈ Z

¯ d+1 satisfy, for each k = (k, kd+1) ∈ Z ,

d Tk¯u(x) − u(x) > 0 or ≡ 0 or < 0 ∀ x ∈ R .

Geometrically, the graph of such a hypersurface when projected on a torus either never intersects itself or overlaps completely if it does intersect itself.

In our setting, it is proved in [11] (Theorem 2.1) that for every nonself-intersecting minimal solution (or minimizer) u to the variational problem, there exists a unique vector ω ∈ Rd, called a rotation vector, such that |u(x) − ω.x| is bounded for all x ∈ Rd. The set of all such minimal solutions corresponding to ω is denoted by M(ω). Under certain conditions on the variational integral ([11], p.243), it is proved

76 that for every ω ∈ Rd M(ω) 6= ∅ (Theorem 5.6), showing the existence of a minimal solution corresponding to ω without self-intersection. Analogously, but in more gen- eral way, we are interested on finding a solution to the variational problem (6.0.1) in the following class of functions

d ∞ d Bω = {u : R → R : u is Birkhoff, u(x) − ω.x ∈ L (R )}

because such a solution is an analogue of a leaf of a minimal foliation or a leaf of

minimal lamination for a variational problem associated to a PDE. We are going to

discuss about such a foliation in the next section.

In [3], Bω is characterized as u ∈ Bω if and only if

u(x + k) − u(x) − kd+1 ≥ 0 if ω.k − kd+1 ≥ 0

u(x + k) − u(x) − kd+1 ≤ 0 if ω.k − kd+1 ≤ 0

¯ d+1 for all k = (k, kd+1) ∈ Z and also mentioned that Bω is closed under several natural topologies, e.g. L2−convergence. We also notice that such a class is invariant under the action of the group Zd+1.

6.2 A Generalization of Minimal Foliations (or Min-

imal Laminations) & Invariant Tori (or Aubry-

Mather Sets)

As we briefly mentioned in the end of Chapter 1, we are interested in studying so- lutions to the variational problem (6.0.1) in the class Bω, as motivated by the stability problem for dynamical systems and the problem of finding minimal foliations for the

77 variational problems associated to PDEs. More specifically, the stability problem for dynamical systems is related to the construction of two-dimensional invariant tori in

KAM theory. The break-down of stability for such systems is related to the disinte- gration of invariant tori. An important work [14] of J. Mather for discrete dynamical systems generated by iterates of area-preserving maps, called monotone twist maps, has analyzed this behavior. Similar work of Aubrey has discussed minimum energy configurations related to the models for the motion of electrons in a one-dimensional crystal. The orbits on invariant sets for monotone twist maps are the analogue of these minimum energy configurations. The stability behavior of two-dimensional twist maps or one-dimensional crystals are related to Hamiltonian systems with two degree of freedom.

Replacing one-dimensional orbits of Hamiltonian systems by hypersurfaces (or manifolds of codimension one) results to the leaves of minimal foliations or a minimal laminations for the variational problems associated to PDEs as a higher dimensional analogue of Aubrey-Mather theory. These are minimal in the sense that their leaves are the graphs of minimal solutions of corresponding variational problems. Let us recall that a smooth foliation of a torus Td is a continuous one parameter family of smooth hypersurfaces, called leaves, that cover Td whereas a (smooth) lamination is a (smooth) foliation of a subset of Td (usually this subset is a Cantor set). According to J. Moser in [13], a Zd+1-invariant F -minimal foliation given by the collection of leaves {xd+1 = u(x, λ): λ ∈ R} corresponding to the variational problem

Z F (x, u, ∇u) dx, Rd where F is periodic of period 1 in variables x and u, has appropriate smoothness and satisfy appropriate growth conditions, is defined by a continuous function u :

Rd × R → R with the following properties

78 1. for each λ ∈ R, u = (x, λ), x ∈ Rd is a solution to the Euler-Lagrange equation, called an extremal of the problem,

2. for each x ∈ Rd, the map λ 7→ u(x, λ) is a homeomorphism of R onto R, and

3. the collection of leaves {xd+1 = u(x, λ): λ ∈ R} is invariant under the action of

the fundamental group Zd+1.

In the case of foliations these extremals u are precisely the minimals to the above

1 d variational problem, where the minimality is defined in the sense: u ∈ Hloc(R ) and

Z F (x, u + φ, ∇u + ∇φ) − F (x, u, ∇u) dx ≥ 0 Rd

1 d for all φ ∈ H0(R ). Indeed, the foliations related to PDEs are the analogues of the invariant tori whereas laminations are the analogues of Aubrey-Mather sets.

Now we will give a couple of examples from [13].

Example 6.2.1. Let us consider a system

d2x = V (t, x) (6.2.1) dt2 x where V (t, x) is smooth and is periodic in t and x with the period 1 in each. We say this system is stable if

sup |x˙(t)| < ∞ t∈R Now we will prove that every such system is stable.

Proof. The system (6.2.1) is the Euler-Lagrange equation of the variational problem

Z 1 E(x) = [ |x˙|2 + V (t, x)] dt (6.2.2) 2

79 Also the system (6.2.1) can be expressed as the Hamiltonian system

1 x˙ = H ;y ˙ = −H , where H(t, x, y) = y2 − V (t, x) (6.2.3) y x 2

For a solution x(t) of it, y = y(t) =x ˙(t) is bounded for all t if it can be enclosed between two invariant tori given by the periodic functions y = w1(t, x) and y = w2(t, x) of period 1 in each t and x. These tori are invariant if the vector field (6.2.3) in T2 × R is tangential to them and thus each such torus can be viewed as a set of orbits. Let us write wi’s, i = 1, 2 as the x-derivative of a function S

y = Sx(t, x). (6.2.4)

Then a necessary and sufficient condition for (6.2.4) to define an invariant torus is that it satisfies the Hamilton-Jacobi equation

St + H(t, x, Sx) = 0 (6.2.5)

where Sx(t, x) has period 1 in each t and x. A solution to the boundary value problem (6.2.5) gives us an invariant torus. The projection of the phase-space T2 × R on T2 takes the orbits on the invariant torus into a family of curves on the torus T2. These orbits are the solutions to the system

x˙ = Sx(t, x), t˙ = 1 (6.2.6)

As these solutions are also solutions of (6.2.1), they form a minimal foliation of the

torus. Conversely, any minimal foliation given by the system (6.2.6) on the torus

defines an invariant torus in T2 × R.

Example 6.2.2. A minimal foliation for a variational problem given by the Dirichlet

80 integral Z |∇u|2 dx (6.2.7) is given by

d u(x) = ω · x + β for some ω ∈ R , β ∈ R.

Proof. The Euler-Lagrange equation of (6.2.7) is the Lapalace equation

∆u = 0

Thus any harmonic function u is a minimal solution to the variational problem (6.2.7).

It will be nonselfintersecting if and only if it has the form

u(x) = ω · x + β for some ω ∈ Rd and β ∈ R. Hence the coninuous family {ω · x + β : β ∈ R} gives rise to a minimal foliation for (6.2.7).

Our study of solutions to PDEs and solutions to ΨDEs in the class Bω generalizes the concepts of leaves of a minimal foliation or a minimal lamination for PDEs and invariant tori or Aubrey-Mathe sets for ODEs. The reason is that those solutions we find in the class Bω are the solutions to Euler-Lagrange equations for the cor- responding varational problem, satisfy nonself-intersection property and also satisfy that their graphs have bounded distance from the hyperplane ud+1 = ω · x. These solutions are less restrictive in the sense that they may be minimal solutions to the corresponding variational problem, which we believe, but have not yet proved in this thesis.

81 6.3 The Proof of Theorem 2.7.5

Now let us restate Theorem 2.7.5 stated in section 2.7.3 as follows:

d+1 ij ∞ d Theorem 6.3.1. Let f ∈ C(T , R) with fy is continuous and a ∈ C (T , R).

d α Then for every ω ∈ Q , there exists u ∈ Bω satisfying (2.1.1), that is A u + f(x, u) = 0.

The case for α = 1 has been proved in [2]. Also the case for Aα = (−∆)α, α ∈

(0, 1) has been proved in [3]. We first prove the case for rational rotation vector,

d d 1 d meaning ω ∈ Q . As ω ∈ Q , we can find N ∈ N such that ω = N k where k ∈ Z . Then we make the conjecture for the case when ω is an irrational rotation vector as a limit of solutions corresponding to rational rotation vectors.

The proof of Theorem 6.3.1 will follow form the next two lemmas.

Lemma 6.3.2. If u(t, x) solves (2.7.1) or (2.7.2) with the initial condition u(0,.) = u0 ∈ Bω, then u(t, .) ∈ Bω for all t > 0.

Proof. We first prove the result related to (2.7.1). Then result related to (2.7.1) follows similarly.

Define operators

d Xkv(x) = v(x + k),Ylv(x) = v(x) + l ∀ k ∈ Z , ∀ l ∈ Z where v : Rd → R.

Since (I + Aα)1−βl = l (see Proof of proposition 4.4.2) and f is Zd+1−periodic, for each fixed t ≥ 0, we have

α 1−β L(Xku(t, x)) = − (I + A ) u(t, x + k)

α 1−β =Xk(−(I + A ) u(t, x))

=Xk(Lu(t, x))

82 and

Xk(Nu(t, x)) =N(u(t, x + k))

=(I + Aα)−β[u(t, x + k) − f(x + k, u(t, x + k))]

α −β =(I + A ) [Xku(t, x) − f(x, Xku(t, x))]

=N(Xku(t, x)).

From the last two equations we obtain that

∂ ∂ (X u(t, x)) = u(t, x + k) ∂t k ∂t =Lu(t, x + k) + Nu(t, x + k)

=Xk(Lu(t, x)) + Xk(Nu(t, x))

=L(Xku(t, x)) + N(Xku(t, x))

This shows that Xku(t, x) is a solution to (2.7.2) with initial condition Xku(0, x) = u(0, x + k) = u0(x + k) For any solution v(t, x) of (2.7.2) corresponding to the initial condition v(0, x) = v0(x), we can define a (nonlinear) solution operator Φt by setting

v(t, x) = Φtv0(x). (6.3.1)

Now replacing v(t, x) by Xku(t, x) and v(0, x) by Xku(0, x) in (6.3.1), we get

ΦtXku(0, x) = Xku(t, x) which yields

ΦtXku0(x) = XkΦtu0(x). (6.3.2)

83 Also we have

α 1−β L(Ylu(t, x)) = − (I + A ) [u(t, x) + l] = Lu(t, x) − l,

α −β N(Ylu(t, x)) =(I + A ) [Ylu(t, x) − f(x, Ylu(t, x)]

=(I + Aα)−β[(u(t, x) + l) − f(x, u(t, x) + l)]

=(I + Aα)−β[u(t, x) − f(x, u(t, x)) + l]

=Nu(t, x) + l

and

∂ ∂ ∂ (Y u(t, x)) = [u(t, x) + l] = u(t, x) = Lu(t, x) + Nu(t, x). ∂t l ∂t ∂t

Therefore, ∂ (Y u(t, x)) = L(Y u(t, x)) + N(Y u(t, x)) ∂t l l l

showing that Ylu(t, x) solves (2.7.2) with the initial condition Ylu(0, x) = Ylu0(x) =

u0(x) + l.

Replacing v(t, x) by Ylu(t, x) and v(0, x) by Ylu(0, x) in (6.3.1), we get ΦtYlu(0, x) =

Ylu(t, x) which follows that

ΦtYlu0(x) = YlΦtu0(x). (6.3.3)

Since u0 ∈ Bω, we have

u0(x + k) − u0(x) − kd+1 ≥ 0 whenever ω.k − kd+1 ≥ 0

or u0(x + k) − u0(x) − kd+1 ≤ 0 whenever ω.k − kd+1 ≤ 0

d+1 for all x and for all (k, kd+1) ∈ Z . But u0(x + k) − u0(x) − kd+1 ≥ 0 implies

84 that XkY−kd+1 u0(x) ≥ u0(x). By the comparison principle (Theorem 2.7.4), we have

ΦtXkY−kd+1 u0(x) ≥ Φtu0(x) and then use (6.3.2) and (6.3.3) in succession to get

XkY−kd+1 Φtu0(x) ≥ Φtu0(x), which is equivalent to

u(t, x + k) − u(x) − kd+1 ≥ 0 whenever ω.k − kd+1 ≥ 0

d+1 for all x and for all (k, kd+1) ∈ Z . Similarly, we have

u(t, x + k) − u(x) − kd+1 ≤ 0 whenever ω.k − kd+1 ≤ 0

d+1 for all x and for all (k, kd+1) ∈ Z . From the last two relations, it follow that

u(t, .) ∈ Bω for all t ≥ 0.

1 d Lemma 6.3.3. Let ω ∈ N k where k ∈ Z ,N ∈ N. If u(t, x) is solution to (2.7.2) d with the initial condition u0(x) = ω.x, x ∈ Ω = NT then there exists a sequence

2 {tn} of times such that tn → ∞ and u(tn,.) → uω in L (Ω) as n → ∞ and uω solves (6.1.1).

Proof. As in Lemma 6.3.2, we prove the result related to (2.7.2) and the result related

to (2.7.1) can be proved similarly.

Writing solution u(t, x) := u[t](x) to (2.7.2) and applying corresponding functional

85 calculus, we have

d d E (u[t]) =DE (u[t]) u[t] dt α α dt  d  = ∇αβEα(u[t]), u[t] dt αβ  d  = (I + Aα)1−βu[t] − (I + Aα)−β(u[t] − f(x, u[t]), u[t] dt αβ  d  = −Lu[t] − Nu[t], u[t] dt αβ  d d  = − u[t], u[t] dt dt αβ 2 d = − u[t] dt αβ ≤0.

This shows that the map t 7→ Eα(u[t]) is a decreasing function of t. Therefore, for every t ≥ 0

Eαu[t] ≤Eαu[0] Z 1 α/2 α/2
= A u0,A u0 L2 + F (., u0) 2 Ω ∞ ∞ ! Z 1 X α/2 X α/2 = λj uˆ0jwj, λk uˆ0kwk + F (., u0) 2 Ω j=1 k=1 L2 ∞ 1 X ≤ λα|uˆ |2 + |Ω| kF k 2 j 0j ∞ j=1 ∞ 1 X ≤ (1 + λα)|uˆ |2 + |Ω| kF k 2 j 0j ∞ j=1

2 = ku0kα + |Ω| kF k∞ .

86 But simplifying the formula for Eα(u[t]), we get

Z α/2 α/2
A u[t],A u[t] L2 = 2{Eαu[t] − F (., u[t])} Ω ∞ Z X α 2 ⇒ λj |uˆj[t]| = 2{Eαu[t] − F (., u[t])} j=1 Ω

P∞ α 2 α P∞ λj α 2 Since 0 < λ1 < λ2 < · · · → ∞, λ |uˆj[t]| = λ ( ) |uˆj[t]| ≥ j=1 j 1 j=1 λ1 α P∞ 2 2α 2α α λ1 j=1 |uˆj[t]| . Since u[t] ∈ H (Ω) and H (Ω) ⊂ H (Ω), we have

∞ 2 X α 2 ku[t]kα = (1 + λj )|uˆj[t]| j=1 ∞ ∞ −α X α 2 X α 2 ≤λ1 λj |uˆj[t]| + λj |uˆj[t]| j=1 j=1 ∞ −α X α 2 =(λ1 + 1) λj |uˆj[t]| j=1 Z −α α =2(λ1 + 1){E u[t] − F (., u[t])} Ω −α 2 ≤2(λ1 + 1)(ku0kα + |Ω| kF k∞ + |Ω| kF k∞)

−α 2 =2(λ1 + 1)(ku0kα + 2|Ω| kF k∞).

α We claim that u0 ∈ H (Ω). To prove the claim, we have

X X X Au = − ∂ (aij(x)∂ (ω.x)) = − ∂ (aij(x)ω ) = − aij (x)ω 0 xj xi xj i xj i i,j i,j i,j

2 P∞ 2 2 which implies that Au0 ∈ L (Ω) and hence j=1 λj |uˆ0j| < ∞. Since 0 < α < 1, P∞ α 2 P∞ 2 2 P∞ 2 2 j=1 λj |uˆ0j| ≤ j=1 λj |uˆ0j| < ∞. Also j=1 |uˆ0j| < ∞ since u0 ∈ L (Ω). Thus

∞ 2 X α 2 ku0kα = (1 + λj )|uˆ0j| < ∞. j=1

−α 2 1/2 Thus ku[t]kα ≤ M for all t ≥ 0, where M = {2(λ1 + 1)(ku0kα + 2|Ω| kF k∞)} .

87 α d α Notice that E (u[t]) is bonded from below. Since dt E (u[t]) ≤ 0, we can choose d α a sequence tn → ∞ as n → ∞ such that dt E (u[tn]) → 0. For instance, if α α we had no such sequence, then we would have 0 ≤ limt→∞ E (u[t]) − E (u[0]) =

R t d α d α limt→∞ 0 ds E (u[s]) ds = a negative number unless dt E (u[t]) ≡ 0 for which case α the result is obvious, showing a contradiction. Since the sequence {u[tn]} in H (Ω) is bounded, by theorem 2.7.4 of [17], the embedding

Hα(Ω) ,→ L2(Ω)

is compact. Therefore u[tn] has a subsequence u[tnk ] that converges to a limit, say

2 2 α 2 uω, in L (Ω). Since A is a closed operator on L (Ω), so is A on L (Ω) (see

α 2 α α [10], [16]). Also A u[tnk ] converges in L (Ω). Thus kA u[tnk ] − A uωkL2 → 0

1 as k → ∞. Since |f(., u[tnk ]) − f(., uω)| ≤ |f|C |u[tnk ] − uω| which implies that

2 1 kf(., u[tnk ]) − f(., uω)kL2 ≤ |f|C ku[tnk ]) − uωkL2 . So under the L −convergence

α α A uω + f(., uω) = lim {A u[tn ] + f(., u[tn ])}. k→∞ k k

Finally, we have

2 d α d − E (u[tnk ]) = u([tnk ]) dt dt αβ

α β/2 2 = (I + A ) {L(u([tnk ])) + N(u([tnk ])) L2

α β/2−β α 2 = (I + A ) {−(I + A )u[tnk ] + u[tnk ] − f(., u[tnk ])} L2

α −β/2 α 2 = (I + A ) {−A u[tnk ] − f(., u[tnk ])} L2 .

88 Letting limit k → ∞ yields

α −β/2 α 2 0 = lim (I + A ) {−A u[tn ] − f(., u[tn ])} 2 k→∞ k k L

α −β/2 α 2 = (I + A ) {−A uω − f(., uω)} L2 since (I + Aα)−β/2 is a closed operator on L2(Ω). Hence

α −β/2 α (I + A ) {−A uω − f(., uω)} = 0

α ⇒ − A uω − f(., uω) = 0 completing the proof.

Notice that the existence of unique solution u(t, x) to (2.7.2) with initial condi- tion u0(x) = ω.x is guaranteed by Theorem 2.7.1 and u(t, .) ∈ Bω for all t ≥ 0 by

2 Lemma 6.3.2 since u0 ∈ Bω. Since Bω is closed under L −convergence, Lemma 6.3.3 guarantees that uω ∈ Bω and uω solves (6.1.1). Moreover, by the regularity theory of

2α pseudo-differential operator, uω ∈ H (Ω). Hence Theorem 6.3.1 is proved for rational rotation vector ω.

Next we prove under the stronger hypothesis than the hypothesis of Theorem

6.3.1 that the solution we have found above is a minimal solution to the variational problem (6.0.1) in the following sense.

α d Definition 6.3.4. A function u ∈ Hloc(R ) is called a minimal solution to the vari- ational problem (6.0.1) if

Eα(u + η) − Eα(u) ≥ 0

α d for all η ∈ H0 (R ).

Theorem 6.3.5. Let f ∈ C(Td, R), and aij ∈ C∞(Td, R). Then for every ω ∈ Qd, there exists u ∈ Bω such that u is a minimal solution to (6.0.1).

89 Proof. Let u ∈ Bω be a solution to (6.1.1). Following the similar reasoning of the

R y d proof of Lamma (2.3.1) with F (x, y) = 0 f(x) dz = yf(x) for x ∈ T and y ∈ R, for α d any η ∈ H0 (R ) we have

Z 1 E (u + η) = |Aα/2(u + η)|2 + F (x, u + η) α 2 Z 1 = |Aα/2u + Aα/2η|2 + (u + η)f 2 Z 1 1 = |Aα/2u|2 + Aα/2uAα/2η + |Aα/2η|2 + uf + fη 2 2 1 Z =E (u) + (Aαu + f, η) + |Aα/2η|2 α L2 2 1 Z =E (u) + |Aα/2η|2 α 2 and hence we obtain

Eα(u + η) − Eα(u) ≥ 0

α d for all η ∈ H0 (R ). This shows that u is a minimal solution to (6.0.1).

From the last theorem, we notice that any solution to the linear pseudo-differential

(Poisson type) equation

Aαu + f = 0 is a minimal solution to the variational problem

Z 1 |Aα/2u|2 + uf 2

Corollary 6.3.6. The variational problem

Z |Aα/2u|2

has a minimal solution in Bω.

90 6.4 Regularity in Some Special Cases

From Chapter 1 in [26], Sobolev embedding theorems for fractional order Sobolev spaces give us the following continuous embeddings.

Hs(Ω) ⊂ Ht(Ω) whenever 0 ≤ t ≤ s < ∞ (6.4.1)

n Hs(Ω) ⊂ Ck(Ω)¯ whenever 0 ≤ k < s − (6.4.2) 2 where k is a nonnegative integer and Ω may be Rd or an open bounded domain. 1 (i) When 2 < α < 1, n = 2 and k = 0, (6.4.2) gives

H2α(Ω) ⊂ C0(Ω¯ ) (6.4.3)

By the above embedding (6.4.3), the solution u to the Euler-Lagrange equation (6.1.1) obtained in Theorem 6.3.1 is continuous on the fundamental domain Ω.

3 (ii) When 4 < α < 1, n = 3 and k = 0, (6.4.2) again gives

H2α(Ω) ⊂ C0(Ω¯ ) (6.4.4) and hence the solution u to the Euler-Lagrange equation obtained in Theorem 6.3.1 is continuous on Ω.

In either of these two cases, graph of hypersurface given by a minimal solution u to the variational problem (6.0.1) given by Theorem 6.3.5 thus represents a leaf of a minimal foliation or a minimal lamination.

91 6.5 Numerical Solutions to Some Semilinear PDEs

and ΨDEs

In this section we will find numerical solutions to some 2D-semilinear partial and pseudo-differential equations involving the Laplacian and fractional powers of the Laplacian under the periodic boundary conditions. We solve these equations by finding limits of the solutions to the corresponding L2(Ω)-gradient equations or

Hαβ(Ω)-gradient equations as t → ∞.

To solve such equations we use discrete Fourier transform and discrete inverse

Fourier transform. First we discretize the fundamental domain Ω = NTd, in most cases the torus T2, into m2 points by placing equally spaced m points on the interval [0,N] including the end points in each of x and y directions. Then we discretize

2 the corresponding frequency domain into m points, m points in each of ξ1 and ξ2 directions. Then we apply Fast Fourier Algorithm (FFT) to compute discrete Fourier transform and Inverse Fast Fourier Algorithm (IFFT) to compute discrete inverse

Fourier transform. We do all these computations and many other numerical compu- tations by developing MATLAB programming codes, which are listed in Appendix

B.

In each of the following examples, ∆t or dt will denote the length of each time step and n denotes the nth time step. While applying DFT, we adapt implicit- explicit method, meaning we calculate the linear term of the equation implicitly and the nonlinear term explicitly.

Example 6.5.1. Now we numerically solve the 2D-second order semilinear partial differential equation

∆v = 2π sin(2π(x + y)) cos(2πv) (6.5.1)

subject to the boundary conditions v(x+ei) = v(x), where ei, i = 1, 2 are the standard

92 2 unit vectors in R , to find a solution in Bω.

We see that the equation (6.5.1) is the Euler-Lagrange equation to the energy functional

Z 1 E(v) = |∇v(x, y)|2 + sin(2π(x + y)) sin(2πv(x, y)) dxdy T2 2 whose L2(Ω)-gradient descent equation is

∂v = ∆v(t, x, y) − 2π sin(2π(x + y)) cos(2πv(t, x, y)). (6.5.2) ∂t

We first find a solution to this equation in Bω subject to the initial condition

v(0, x, y) = v0(x) = ω.x ∈ Bω where ω ∈ Z2 and the boundary conditions

v(t, x + 1, y) = v(t, x, y) = v(t, x, y + 1).

Then find the limit of the solution as t becomes large enough so that the limit will satisfy the equation (6.5.1) (according to Lemma 6.3.3, however we will justify nu- merically).

Applying DFT on the equation (6.5.2) with forward-backward Euler (implicit- explicit) method, we obtain

(\vn+1) − (dvn) k k = (|ik|2)(\vn+1) − {(2π sin(2π(x\+ y)) cos(2πv))n} ∆t k k

2 2 n+1 n n or {1 − ∆t(−k1 − k2 )}(\v )k = (dv )k − ∆t{(2π sin(2π(x\+ y)) cos(2πv)) }k n \ n n+1 (dv )k − ∆t{(2π sin(2π(x + y)) cos(2πv)) }k or (\v )k = 2 2 , 1 − ∆t(−k1 − k2 )

93 m m n where k = (k1, k2), k1, k2 = − 2 + 1,..., 2 and v (., .) = v(n∆t, ., .).

0 Using the initial time step solution (vb )k = (vb0)k in the last formula, we evaluate 1 1 1 1 2 2 (vb )k and then v as IFFT of vb . Using (vb )k, we evaluate (vb )k and then v as IFFT

of vb2 and so on. As time keeps evolving, that is, as the nth time step goes forward and n becomes large enough, we will see from MATLAB simulation that the graph of the

solution v to the initial-boundary value problem eventually remains stationary. Thus

this v with the stationary surface for large enough n will satisfy the boundary value

problem (6.5.1) with the significant degree of accuracy. In our calculations we have

adapted the accuracy at least to the four figures of decimal. The surface represented

by the solution v to the boundary value problem (6.5.1) are given with different orien-

tations in Figure (a) and Figure (b). We can also notice from these figures that v ∈ Bω.

Example 6.5.2. We numerically find a solution in Bω to the semilinear pseudo- differential equation

(−∆)αv + 2π sin(2π(x + y)) cos(2πv) = 0 (6.5.3)

subject to the periodic boundary conditions as mentioned in Example 6.5.1.

We can easily see that the equation (6.5.3) is the Euler-Lagrange equation to the 94 functional Z 1 α/2 2 Eα(u) = |(−∆) | + sin(2π(x + y)) sin(2πv) T2 2 whose L2(Ω)-gradient descent equation is

∂v = −(−∆)αv(t, x, y) − 2π sin(2π(x + y)) cos(2πv(t, x, y)). (6.5.4) ∂t

We impose the same initial and boundary conditions as in Example 6.5.1.

We apply DFT on the equation (6.5.4) with forward-backward Euler method to

obtain

\n+1 n (v )k − (dv )k 2 α n+1 n = −(−|ik| ) vd k − ({2π sin(2π(x\+ y)) cos(2πv)} ) ∆t k

2 2 α n+1 n n ⇒ {1 + ∆t(k1 + k2) }(\v )k = (dv )k − ∆t({2π sin(2π(x\+ y)) cos(2πv)} )k (dvn) − ∆t({2π sin(2π(x\+ y)) cos(2πv)}n) \n+1 k k ⇒ (v )k = 2 2 α . 1 + ∆t(k1 + k2)

We present this formula in MATLAB programming code in Appendix B. As time

evolves, the surface represented by the solution obtained from the numerical simula-

tion remains stationary for large enough n and is shown in Figure (c) through Figure

(g) for different values of α.

The error in the second column of Table 1 refers to the approximation of (−∆)αv+

2π sin(2π(x+y)) cos(2πv). We can notice from Figure 3 and the first row of Table (h) that v is not a solution to the given boundary value problem for α = 0.2. Moreover, from Figure (d) through Figure (g) and the remaining rows of of Table (h) we can conclude that v is a solution to the given boundary value problem when α is greater than 0.5 and increases to 1. These arguments somehow justify the assertion made in section 6.4.

More interestingly, what we notice from Figure (d) through Figure (g) that as α approaches 1 from the left, the solution to the boundary value problem (6.5.3) ap-

95 proaches the solution to the boundary value problem (6.5.1). This can be seen from how the surface represented by the solution to the boundary value problem (6.5.3) is

96 becoming closer and closer to the surface represented by the solution to the boundary

value problem (6.5.1) as α approaches 1.

Example 6.5.3. Finally we numerically solve the boundary value problem in Example

6.5.2 by the help Sobolev gradient descent equation.

We can notice that the Hαβ(Ω)-gradient descent equation corresponding to the

equation (6.5.3) is

∂v = −(I + (−∆)α)1−βv + (I + (−∆)α)−β(v − 2π sin(2π(x + y)) cos(2πv)). (6.5.5) ∂t

We consider the same initial and boundary conditions as in Example 6.5.1. Applying

DFT on this equation, we obtain

(\vn+1) − (dvn) k k = − (1 + |k|2α)1−β(\vn+1) ∆t k

2α −β n n + (1 + |k| ) [(dv )k − ({2π sin(2π(x\+ y)) cos(2πv)} )k]

2α 1−β n+1 n or {1 + ∆t(1 + |k| ) }(\v )k = (dv )k

2α −β n n + ∆t(1 + |k| ) [(dv )k − ({2π sin(2π(x\+ y)) cos(2πv)} )k]

(dvn) + ∆t{1 + (k2 + k2)α}−β[(dvn) − ({2π sin(2π(x\+ y)) cos(2πv)}n) ] \n+1 k 1 2 k k or (v )k = 2 2 α 1−β 1 + ∆t{1 + (k1 + k2) }

Its corresponding MATLAB programming code is presented in Appendix B and the resulting solution to the boundary value problem (6.5.3) is shown in Figure 8 through

Figure 13 for different values of α and β.

In the above figures, we can clearly notice that for each fixed value of α the sur- faces remain unchanged when β changes from 0.1 to 0.2. For instance, the surfaces

97 in Figure 8 and Figure 9 look the same and match with the surface in Figure 5.

Similarly, each of the surfaces in Figure 10 and Figure 11 matches with the surface

98 in Figure 6 and each of the surfaces in Figure 12 and Figure 13 matches with the surface in Figure 7.

Hence we can conclude that the solutions to the L2(Ω)-gradient descent equation

αβ and the H (Ω)-gradient descent equation approach a same function, say vω(x), as t becomes large enough and that the limit function vω satisfies the boundary value problem (6.5.3).

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102 Appendix A

Pseudo-differential Operators

Here we introduce some terms and useful results from pseudo-differential operator

theory, functional analysis and distribution theory (ref. [17, 19, 18]).

Definition A.0.4 (The Schwartz space and the space of tempered distribution). The

Schwartz space S(Rd) is the space of functions u ∈ C∞(Rd) such that for any multi- indices α and β

sup |xα(Dβu)(x)| < ∞, x∈Rd in other words, all derivatives u decrease faster than any power of |x| as |x| approaches

∞.

The Space S0(Rd) of tempered distributions is the space of all continuous linear functionals on S(Rd). (e.g. Fourier transform is a tempered distribution).

Definition A.0.5 (The space of distributions). Let X be a domain. The space D0(X)

∞ of distributions is the space of all continuous linear functionals on C0 (X).

∞ ∞ Here the continuity in the sense of C -topology on C0 (X) and the weak topology on D0(X) defined by the seminorms

∞ ρϕ(f) = |hf, ϕi|, ϕ ∈ C0 (X)

The weak topology on D0(X) can be characterized by the following convergence 103 on it:

0 0 A net fλ in D (X) converges in D (X) to a distribution f if

∞ 1) limλhfλ, ϕi = hf, ϕi ∀ ϕ ∈ C0 (X) and 2) for any multiindex α

α α ∞ limh∂ fλ, ϕi = h∂ f, ϕi ∀ ϕ ∈ C0 (X) λ where αth distributional derivative of g ∈ D0(X) is defined by

α |α| α ∞ h∂ g, ϕi = hg, (−1) ∂ ϕi ∀ ϕ ∈ C0 (X)

Let X be an open subset of Rd.Then

m N Definition A.0.6 (The space S%,δ(X × R )). For any real numbers m, %, δ with

0 ≤ %, δ ≤ 1 it consists of all functions a(x, θ) ∈ C∞(X × RN ) such that for any multiindices α and β and any compact subset K of X there is a constant Cα,β,K such that

α β m−%|α|+δ|β| |∂θ ∂x a(x, θ)| ≤ Cα,β,K hθi where hθi2 = 1 + |θ|2

Definition A.0.7 (Pseudo-differential operators, ΨDO). Define the operator A by

ZZ i(x−y).ξ ∞ Au(x) = e a(x, y, ξ)u(y) dydξ, u ∈ C0 (X),

m d ∞ 0 where a(x, y, ξ) ∈ S%,δ(X × X × R ). Then the operator A : C0 (X) → D (X) defined by ZZZ i(x−y).ξ ∞ hAu, vi = e a(x, y, ξ)u(y)v(x) dxdydξ, v ∈ C0 (X) is called a pseudo-differential operator determined with the symbol a(x, y, ξ). The

m d class of all pseudo-differential operators with the symbol a(x, y, ξ) ∈ S%,δ(X × X × R ) 104 m is denoted by L%,δ(X).

Definition A.0.8 (Kernel & properly supported ΨDO). The distribution KA ∈ D0(X × X) defined by

∞ ∞ hKA, u(y)v(x)i = hAu, vi, u ∈ C0 (X), v ∈ C0 (X)

is called the kernel of A. If the projections π1, π2 : supp(KA) → X are proper maps, then A is called a properly supported ΨDO.

For example, consider the linear differential operator

X α A = aα(x)D |α|≤2

∞ 1 1 where aα ∈ C (X) and D = ( i ∂x1 , ...... , i ∂xd ). Using the Fourier transform we have

ZZ Dαu(x) = ei(x−y).ξξαu(y)dydξ and hence ZZ i(x−y).ξ Au(x) = e σA(x, ξ)u(y)dydξ

P α with the symbol σA(x, ξ) = |α|≤2 aα(x)ξ . We notice that A is a properly supported

2 d ΨDO with σA(x, ξ) ∈ S1,0(X × R ) and supp(KA) = {(x, x): x ∈ X}. We have by Proposition 3.1 in [17] If A is a properly defined ΨDO, then it defines a map

∞ ∞ A : C0 (X) → C0 (X) which extends to continuous maps:

A : E 0(X) → E 0(X) 105 A : C∞(X) → C∞(X)

A : D0(X) → D0(X)

Remark: If a(x, y, ξ) is properly supported symbol, meaning the projections

π1, π2 : suppx,ya(x, y, ξ) ⊂ X × X → X are proper maps, then the corresponding ΨDO is properly supported.

α ∞ A linear differential operator A = Σ|α|≤maα(X)D with aα ∈ C (X) is elliptic if

α the principal symbol am(x, ξ) = Σ|α|=maα(x)ξ satisfies

n am(x, ξ) 6= 0, (x, ξ) ∈ X × (R \ 0)

Next theorem is about compact embedding (Theorem 7.4 in [17])

Theorem A.0.9 (Compact embedding). Let s > s0 and K ⊂ M is compact. Then we have the embedding Hs(K) ,→ Hs0 (K).

Now let us take uniformly elliptic selfadjoint operator A : C∞(Ω) → C∞(Ω) with

P α Pn ij the principal symbol a2(x, ξ) = |α|=2 aα(x)ξ = i,j=1 a (x)ξiξj, which has a con-

0 0 tinuous extension A : D (Ω) → D (Ω). Then the operator A0 := A|H2(Ω) : D(A0) ⊂

2 0 2 2 2 2 L (Ω) → D (Ω) has D(A0) = H (Ω) and R(A0) = L (Ω) since A : H (Ω) → L (Ω)

2 is a topological isomorphism. We will write A for A0 as an operator on L (Ω).

2,2 Since A ∈ H1,0 (Ω), the space of hypoelliptic operators (see [17]), by inverse

−1 −1 operator theorem (Theorem 8.2, [17]) (A0 − λI) = (A − λI) is compact in L2(Ω) for λ∈ / σ(A). By theorem 8.3, [17], there is a complete orthonormal sys-

∞ ∞ 2 tem {wj}j=1 (⊂ C (Ω)) ( basis for L (Ω)) of eigenfunctions of A corresponding to

∞ the eigenvalues {λj}j=1 with |λj| → ∞ as j → ∞. Moreover σ(A) = σp(A).

106 Appendix B

MATLAB Programmings

Here we present MATLAB programming codes we have developed for discussing the behavior of solutions to semilinear partial differential equations and semilinear pseudo-differential equations in section 6.5.

Among the programming codes given in next few pages, Code A is for Example

6.5.1, Code B for Example 6.5.2 and Code C is for Example 6.5.3.

B.1 Code for Example 6.5.1

107 Code A

%Solving nonlinear partial differential equation equation: % (-laplacian) v+V_v(v)=0, where V(v)=2*pi*cos(2*pi*(xx+yy))sin(2*pi*v) %by usign Implicit-Explicit method on L^2-gradient descent equation %v_t=-(-laplacian)v-V(v)=laplacian-sin(2*pi*v) %B.C.: Z^2-periodic, i.e. v(t,x+e_i)=v(x),i=1,2 %I.C.:v(0,x)=omega.x clc clear om_x=1; om_y=1; %omega=(om_x,om_y)

%Grid N =256; h = 1/N; x = h*(1:N); dt = .01;

%x and y meshgrid in fundamental domain y=x'; [xx,yy]=meshgrid(x,y);

%Initial conditions v=om_x*xx+om_y*yy;

%(ik) and (ik)^2 in frequency domain in xi_1 & xi_2 directions ikx= (2*pi*1i*[0:N/2-1 0 -N/2+1:-1]); iky= (2*pi*1i*[0:N/2-1 0 -N/2+1:-1]'); k2x=ikx.^2; k2y=iky.^2;

%(ik)^2 meshgrides in xi_1 and xi_2 directions [kxx,kyy]=meshgrid(k2x,k2y); for n = 1:500 n V=2*pi*sin(2*pi*(xx+yy)).*cos(2*pi*(v)); %nonlinear term %FFT for linear and nonlinear terms V_hat = fft2(V); v_hat=fft2(v); % DFT of v at (n+1)th timestep by Implicit-Explicit method v_hatnew=(v_hat-dt*(V_hat))./ ... (1-dt*(kxx+kyy)); v=ifft2(v_hatnew); %converts to v defined on fundamental domain V=2*pi*sin(2*pi*(xx+yy)).*cos(2*pi*(v));% computes for final v

%Plotting each timestep solution mesh(v); title(['Timesteps = ',num2str(n)]); axis([0 N 0 N 0 1.5]); xlabel x; ylabel y; zlabel v; view(40,20); drawnow;

m=N; dx=h; dy=h; v_hat=fft2(v);

%Evaluating partial derivatives in x direction for i=1:m for j=2:m-1 der_v_xx(i,j)=(v(i,j+1)-2*v(i,j)+v(i,j-1))/(dx)^2; end der_v_xx(i,1)=(v(i,2)-2*v(i,1)+v(i,m))/(dx)^2; der_v_xx(i,m)=(v(i,1)-2*v(i,m)+v(i,m-1))/(dx)^2; end

%Evaluating partial derivatives in y direction for j=1:m for i=2:m-1 der_v_yy(i,j)=(v(i+1,j)-2*v(i,j)+v(i-1,j))/(dy)^2; end der_v_yy(1,j)=(v(2,j)-2*v(1,j)+v(m,j))/(dy)^2; der_v_yy(m,j)=(v(1,j)-2*v(m,j)+v(m-1,j))/(dy)^2; end end

Er=max(max(abs(der_v_xx+der_v_yy-V))) %checks accuracy of solution to the original BVP

%%%%% Or following method can aslo be adapted %%%%%%%%%% %A_alphav=ifft2((kxx+kyy).*(v_hat)); %%Computes (lap)v=ift((|ixi|^2)ft(v))

%Er=max(max(abs(A_alphav-V))) %%Computes (lap)v-f(.,v)

Code B

%Solving nonlinear psuedo-differential equation equation %(-laplacian)^(alpha)v+f(v)=0, where f(v)=2*pi*cos(2*pi*(xx+yy))sin(2*pi*v) %by usign Implicit-Explicit method on L^2-gradient descent equation %v_t=-(-lap)^(alpha)v-f(v) %B.C.: Z^2-periodic, i.e. v(t,x+e_i)=v(x),i=1,2 %I.C.:v(0,x)=omega.x clc clear om_x=1; om_y=1; %omega=(om_x,om_y) alpha=0.8;

%Grid N =256; h = 1/N; x = h*(1:N); dt = .01;

%x and y meshgrid in fundamental domain y=x'; [xx,yy]=meshgrid(x,y);

%Initial conditions v=om_x*xx+om_y*yy;

%%(ik),(ik)^2 & (ik)^2 meshgrid in frequency domain ikx= (2*pi*1i*[0:N/2-1 0 -N/2+1:-1]); iky= (2*pi*1i*[0:N/2-1 0 -N/2+1:-1]'); k2x=ikx.^2; k2y=iky.^2; [kxx,kyy]=meshgrid(k2x,k2y); for n = 1:500 n fv= 2*pi*sin(2*pi*(xx+yy)).*cos(2*pi*(v)); fv_hat = fft2(fv); v_hat=fft2(v); v_hatnew=(v_hat-dt*(fv_hat))./ ... (1+dt*(-kxx-kyy).^(alpha));

v=ifft2(v_hatnew); fv= 2*pi*sin(2*pi*(xx+yy)).*cos(2*pi*(v));

%Plotting each timestep solution v mesh(v); title(['alpha = 0.8, timesteps = ',num2str(n)]); axis([0 N 0 N 0 1.5]); xlabel x; ylabel y; zlabel v; view(40,20); drawnow;

m=N; dx=h; dy=h; v_hat=fft2(v); end

%Computing (-lap)^(alpha)v= (- lap)^(alpha)v=ift((|xi|^2)^(alpha)ft(v)) A_alphav=ifft2((-kxx-kyy).^(alpha).*(v_hat));

%Computing (-lap)^(alpha)v+f(v) to check if it approaches 0 Er=max(max(abs(A_alphav+fv)))

Code C

%Solving nonlinear psuedo-differential equation equation %(-laplacian)^(alpha)v+f(v)=0, where f(v)=2*pi*cos(2*pi*(xx+yy))sin(2*pi*v) %by usign Implicit-Explicit method on H^{alphabeta}-gradient descent equation %v_t=-{I+(-lap)^(alpha)}^(1-beta)v+{I+(-lap)^(alpha)}^(-beta)(v- f(v)) %B.C.: Z^2-periodic, i.e. v(t,x+e_i)=v(x),i=1,2 %I.C.:v(0,x)=omega.x clc clear om_x=1; om_y=1; %omega=(om_x,om_y) beta=0.1; alpha=0.8;

%Grid N =256; h = 1/N; x = h*(1:N); dt = .01;

%x and y meshgrid in fundamental domain y=x'; [xx,yy]=meshgrid(x,y);

%Initial conditions v=om_x*xx+om_y*yy;

%%(ik),(ik)^2 & (ik)^2 meshgrid in frequency domain ikx= (2*pi*1i*[0:N/2-1 0 -N/2+1:-1]); iky= (2*pi*1i*[0:N/2-1 0 -N/2+1:-1]'); k2x=ikx.^2; k2y=iky.^2; [kxx,kyy]=meshgrid(k2x,k2y); for n = 1:200 n fv=2*pi*sin(2*pi*(xx+yy)).*cos(2*pi*(v)); fv_hat = fft2(fv); v_hat=fft2(v); num=v_hat+dt*(1+(-kxx-kyy).^(alpha)).^(-beta).*(v_hat-fv_hat); den=1+dt*(1+(-kxx-kyy).^(alpha)).^(1-beta); v_hatnew=num./den;

v=ifft2(v_hatnew); fv= 2*pi*sin(2*pi*(xx+yy)).*cos(2*pi*(v));

%Plotting each timestep solution v mesh(v); title(['alpha = 0.8, beta = 0.1, timesteps = ',num2str(n)]); axis([0 N 0 N 0 1.5]); xlabel x; ylabel y; zlabel v; view(40,20); drawnow;

m=N; dx=h; dy=h; v_hat=fft2(v); end

%Computing (-lap)^(alpha)v= (- lap)^(alpha)v=ift((|xi|^2)^(alpha)ft(v)) A_alphav=ifft2((-kxx-kyy).^(alpha).*(v_hat));

%Computing (-lap)^(alpha)v+f(v) to check if it approaches 0 Er=max(max(abs(A_alphav+fv)))