Regularity and Stability of a Wave Equation with a Strong Damping and Dynamic Boundary Conditions

EVOLUTION EQUATIONS AND doi:10.3934/eect.2013.2.631 CONTROL THEORY Volume 2, Number 4, December 2013 pp. 631–667

REGULARITY AND STABILITY OF A WAVE EQUATION WITH A STRONG DAMPING AND DYNAMIC BOUNDARY CONDITIONS

Nicolas Fourrier

Department of Mathematics, University of Virginia Charlottesville, VA 22904, USA CGG, Massy, France

Irena Lasiecka

Department of Mathematics, University of Memphis Memphis, TN 38152-3370 IBS, Polish Academy of Sciences, Warsaw, Poland

Abstract. We present an analysis of regularity and stability of solutions corresponding to wave equation with dynamic boundary conditions. It has been known since the pioneering work by [26, 27, 30] that addition of dynamics to the boundary may change drastically both regularity and stability properties of the underlying system. We shall investigate these properties in the context of wave equation with the damping aﬀecting either the interior dynamics or the boundary dynamics or both. This leads to a consideration of a wave equation acting on a bounded 3-d domain coupled with another second order dynamics acting on the boundary. The wave equation is equipped with a viscoelastic damping, zero Dirichlet boundary conditions on a portion of the boundary and dynamic boundary conditions. These are general Wentzell type of boundary conditions which describe wave equation oscillating on a tangent manifold of a lower dimension. We shall examine regularity and stability properties of the resulting system -as a function of strength and location of the dissipation. Properties such as well- posedness of ﬁnite energy solutions, analyticity of the associated semigroup, strong and uniform stability will be discussed. The results obtained analytically are illustrated by numerical analysis. The latter shows the impact of various types of dissipation on the spectrum of the generator as well as the dynamic behavior of the solution on a rectangular domain.

1. Introduction. We consider the following damped wave equation with dynamic boundary conditions:

2010 Mathematics Subject Classiﬁcation. Primary: 35Lxx, 35Kxx, 35Pxx; Secondary: 65kxx, 76Nxx. Key words and phrases. Wave equation with dynamic boundary conditions, strong damping, Laplace’a Beltrami operator, semigroup generation, analyticity of semigroups, Gevrey’s regularity, strong stability, uniform stability, spectral analysis. The second author is supported by NSF grant DMS-0104305 and AFOSR Grant FA9550-09-1- 0459.

631 632 NICOLAS FOURRIER AND IRENA LASIECKA

u k ∆u ∆u = 0 in Ω tt − Ω t − u ∂ (u + k u ) k ∆ (αu + u) = 0 in Γ  tt ∂ν Ω t Γ Γ t 1 (1) u =− 0 − in Γ  0  u(0, x) = u0, ut(0, x) = u1 on Ω  where u = u(x, t), t 0, x Ω is a bounded three dimensional domain, ∆ denotes ≥ ∈ the Laplacian operator with repect to the x variable, δΩ = Γ0 Γ1,Γ0 Γ1 = , ∂ ∪ ∩ ∅ ∂ν denotes the outer normal derivative, ∆Γ denotes the Laplace-Beltrami operator on the boundary Γ1 with repect the variable x, kΩ, kΓ and α are non-negative constant. In order to ensure the presence of damping and thus energy dissipation in this system, we impose the following condition:

max kΩ, kΓα > 0 (2) { }

The term ∆ut which we will call the viscoelastic damping in the interior indicates that the stress is proportional not only to the strain but also the strain rate. This equation corresponds to a physical model describing gas dynamics evolving in a bounded three dimensional domain Ω with a boundary denoted by Γ, and it was first described by Morse-Ingrad in [34]. More precisely, we consider gas undergoing small irrotational perturbations away from Ω which are concentrated on the portion of the boundary denoted by Γ1. In general, when a small-amplitude plane wave hits a plane surface such as a wall, some of the energy is absorbed by the surface while some is reflected. The pressure generated, during this phenomenon, will tend to make the surface move, or else tends to force more fluid into the pores of the surface, creating, in the tangeantial direction motion, which will be characterized by the surface properties. When the motion at one point is independant to the motion of any other part of the surface; thus each point acts like a spring in response to the excess pressure in the gas, such a surface is called one of local reaction. However, if the surface behavior at one point depends on the behavior at neighboring points, so that the reaction is different for different incident waves, the surface can be called one of extended reaction. It means that the boundary condition are more complex that a simple harmonic oscillator, i.e., the acceleration is not proportional to the displacement anymore but to the relative displacement compared to its neighbours. Such surfaces are of many sorts: one which behaves like membranes and one with laminated structure, in which waves are propagated parallel to the surface; one in which waves penetrate into the material of the wall. From the mathematical point of view, these problems do not neglect acceleration terms on the boundary. Such type of boundary conditions are usually called dynamic boundary conditions and find numerous of applications in the bio-medical domain [10, 43], as well as in applications related to stabilization and active control of large elastic structures. See [33] for some applications. While undamped wave equations are well-posed in an Lp-setting if and only if p = 2 or the space dimesion is 1 ([25]), it is well-known that the wave equation with viscoelastic damping - strongly damped wave equation -, i.e., kΩ > 0, under classical boundary conditions such as Dirichlet or Robin generates an exponentially stable analytic C0-semigroup [11]. In fact, this result is known for a larger class of scalar problems where viscoelastic damping is given in a form of fractional powers of Laplacian. STRONG DAMPED WAVE EQUATION WITH DBC 633

The purpose of this paper is to study how the dynamic boundary conditions on Γ1 aﬀect the properties of the dynamics. We are particularly interested in quali- tative properties of the resulting semigroup that include regularity and long time behavior with the analysis of various types of stability. It is well known that the presence of a damping impacts these two properties. Suﬃcient amount of dissipation in the system may provide not only strong decays of the energy when time t , but also may regularize semigroup by providing more smoothness to the solutions.→ ∞ It is also known that “more damping” does not mean more decays. The phenomenon of overdamping is well known both in practice (engineering) and in theory (mathematics). This motivates our interest in studying the balance between the competing damping mechanisms in the interior Ω and on the boundary Γ. This general problem is, of course, not new and goes back to the fundamental work by Littman-Markus [26, 27, 29] where “hybrid ” systems of elasticity were studied. It was then discovered that an addition of a lower dimensional dynamics may destabi- lize the system. In a similar vein it was shown in [30] that the hybrid system with kΩ > 0

utt ∆u = 0 x Ω, t > 0 u =− 0 x ∈ Γ , t > 0  ∈ 0 mu + ∂ (u + u ) = 0 x Γ , t > 0  tt ∂ν t 1  u(0, x) H1(Ω), u (0, x) L (Ω) x ∈ Ω ∈ t ∈ 2 ∈  with m > 0 is not uniformly stable, while the case when m = 0 is exponentially stable assuming suitable geometric conditions imposed on Ω [24]. The ﬁrst mathematical model for surface of local reaction was introduced by Beale-Rosencrans, in [5], where he described the acoustic/structure interaction by a wave equation with acoustic boundary conditions:

u ∆u = 0 x Ω, t > 0 tt − ∈ u(x, t) = 0 x Γ0, t > 0  ∂ ∈  m(x)δtt + +d(x)δt + k(x)δ = ρ ∂ν ut x Γ1, t > 0  ∂ − ∈  δt = ∂ν u x Γ1  1 ∈  u(0, x) H (Ω), ut(0, x) L2(Ω) x Ω z(0, x) ∈ H1(Ω), z (0, x) ∈L (Ω) x ∈ Ω ∈ t ∈ 2 ∈     Beale, in [7,6] demonstrated that the problem was governed by a C0-semigroup of contractions. Later, similar models were studied ([20] and references therein) pointing out long time behavior and continuous dependance of solutions on the mass of the structure. Also, an extensive study of this model within an abstarct framework is given by Mugnolo in [35]. Not only acoustic boundary conditions and the present study model similar physical phenomenon but they also coincide with the so-called general Wentzell boundary conditions (GWBC) given conditions on the parameters ([16]). Such boundary conditions involve second derivatives as well as lower order terms (Robin type) and Laplace-Beltrami terms and were intensively studied by Favini, Goldstein, Gal et al., in [12, 14] and references therein, in the context of hybrid problems. The Wentzell-type boundary conditions are intrinsically related to the one in our model (1), as we will see in section7. Other papers used this framework in the study of one-dimensional wave equation without internal damping, with Wentzell boundary conditions, see [1, 13, 44]. 634 NICOLAS FOURRIER AND IRENA LASIECKA

Wave equations with viscoelastic damping and dynamic boundary conditions acting on a surface of local reaction have been studied within the framework of the following general model:

utt kΩ∆ut ∆u = f1(u) x Ω, t > 0 u =− 0 − x ∈ Γ , t > 0  ∈ 0 (3) u + ∂ (u + k u ) + ρ(u ) + f (u) = 0 x Γ , t > 0  tt ∂ν Ω t t 2 1  u(0, x) = u , u (0, x) = u x ∈ Ω 0 t 1 ∈ Gerbi and Said-Houari in [18, 17] studied the problem (3) with f2 = 0, f1(u) = p−2  m−2 u u and a nonlinear boundary damping term of the form ρ(ut) = ut . A| | local existence result was obtained by combining the Faedo-Galerkin| method| with the contraction mapping theorem. The authors also showed that under some restrictions on the exponents m and p, there exists inital data such that the solution is global in time and decay exponentially. Graber and Said-Houari in [19] extended some of these results to more general function f1 and f2. They also demonstrate that for ρ = f2 = 0, the model (3) was governed by an analytic semigroup on 1 2 2 H (Ω) L (Ω) L (Γ1). Γ0 × × Remark 1. We note that the model under consideration (1 ) can be recast in the abstract form as u = Au(t) + Cu (t) tt t (4) w = B u(t) + B u (t) + B u(t) + B u (t) tt 1 2 t 3 4 t where the operators A, C, Bi, i = 1 4 satisfy suitable conditions and variables u and w are connected via “trace” type− operator L so that w = Lu. In fact, this is the framework pursued in by Mugnolo in [36] where the second order abstract system (4) is reformulated as an abstract second order Cauchy problem on the product space X ∂X in the variable U(t) (u(t), Lu(t)) satisfying × ≡ Utt = U(t) + Ut(t), t > 0 (5) A C on a product space = X ∂X where X × A 0 C 0 , A ≡ B1 B3 C ≡ B2 B4 are operator matrices on with suitably defined domains. We also mentionned that a similar approach was usedX by Xiao et al. in [46, 45], where boundary conditions were reformulated as differential inclusions in suitable equivalence classes. However, the authors only discussed smoothing properties, without explicit generation result of analytic semigroups. The above formulation can be further reduced to first order Cauchy problem by introducing variable u (U, U ) and writing ≡ t

ut(t) = Au(t) 0 I A ≡ AC A comprehensive study of well-posedness and regularity of second order evolutions (5) is given in [36] under the following standing set of general assumptions: 1. Y, X, ∂Y, ∂X are Banach spaces such that Y, X, ∂Y , ∂X. 2. A : D(A) X X, C : D(C) X X are→ linear opertaors.→ 3. L : D(A) ⊂D(C→) ∂X is linear⊂ and→ surjective. ∩ → STRONG DAMPED WAVE EQUATION WITH DBC 635

4. B : D(A) ∂X, B : D(C) ∂X are linear operators. 1 → 2 → 5. B3 : D(B3) ∂Y ∂X, B4 : D(B4) ∂X ∂X, are linear closed operators. ⊂ → ⊂ → Thus, in order to compare our results with these obtained in [36] we shall recast our problem within this more general framework. This is easily accomplished by setting: A = ∆,C = k ∆, Lu = γ u , Ω ≡ |Γ1 ∂ ∂ B = ,B = k ,B = k ∆ ,B = k α∆ 1 −∂ν 2 − Ω ∂ν 3 Γ Γ 4 Γ Γ with the corresponding spaces (assuming kΓ > 0 ): 1 1 X = L2(Ω),Y = HΓ0 (Ω), ∂X = L2(Γ1), ∂Y = H (Γ1)

The basic framework presented in [36] aims at proving generation of C0 semigroups (resp. analyticity) as being equivalent having the same properties for the blocks of operators 0 I 0 I , and AC B B 3 4 considered on Y X and ∂Y ∂X. (see Thm 3.3, 3.8, 4.5, 4.12 [36]). Clearly the role× of the “coupling”× operator L is critical. In fact, the results in [36] are categorized with respect to the properties of L as unbounded Y ∂X , • bounded from →Y ∂X but unbounded X ∂X. • → → It is this second scenario that is relevant in our situation. The corresponding results are in section 4 [36]. As we shall see below the results of both Theorem 4.5 and Theorem 4.12 are not applicable -due to severity of assumptions imposed either on the operators Bi, i = 1 4. To wit, Part I of Thm 4.5 assumes that B1 − ∂ ∈ L(Y, ∂X),B2 L(X, ∂X). The above is never satisfied with B1 = ∂ν ,B2 = ∂ ∈ − kΩ ∂ν and the choices of spaces X, Y, ∂X. In addition, operators B3,B4 do not −comply with regularity requirements postulated in Thm 4.5-unless they are bounded. Similar conclusion applies to Part II of Thm 4.5. Part II of that Theorem pertains to generation of analytic semigroups. However, here the operators B1 and B2 do not comply with regularity requirements unless the Dirichlet map is sufficiently smooth- 1/2 as in one dimensional case. (note that (D(A)L) corrresponds to H (Ω) , unless the dimension of Ω = 1. The main reason is that the treatment given in [36] treats the coupling operator B1,B2 like a perturbation -rather than a carrier of regularity. It is this second approach that is used in our paper where the matrix operators is not a perturbation of two blocks of operator matrices -but rather perturbation of a new system which is related to Wentzell problem. What distinguishes our model from other systems studied is the presence of Laplace Beltrami operator in the dynamic boundary condition. The latter trans- forms the local reaction surface into an extended reaction surface. In the case of viscoelastic dampers located in Ω, our aim is to provide a comprehensive study of (i) well-posedness, (ii) regularity and (iii) stability of solutions under the influence of competing both interior and boundary damping. In order to quantize the analysis we introduce the energy function. 636 NICOLAS FOURRIER AND IRENA LASIECKA

With respect to this energy we are interested in the well-posedness and stability T of solutions (u|Ω, ut|Ω, u|Γ1 , ut|Γ1 ) to (1), along with the decay of the energy E(t) of (1) at t .We begin by deﬁning energy functions representing both internal and boundary→ ∞ energy of the system as well as the damping term D(t):

E(t) = EΩ(t) + EΓ(t)

E (t) = u(t) 2 + u (t) 2 dΩ Ω |∇ | | t | ZΩ E (t) = k u (t) 2 + u (t) 2 dΓ Γ1 Γ|∇Γ |Γ1 | | t|Γ1 | 1 ZΓ1 D(t) = k u (t) 2dΩ + k α u (t) 2dΓ Ω|∇ t | Γ |∇Γ t|Γ1 | 1 ZΩ ZΓ1 We have the following (formal) energy identity:

E(0) = EΩ(t) + EΓ1 (t) + 2 D(s)ds Z0 2 2 2 2 E(0) = u(t) Ω + ut(t) Ω + kΓ Γu|Γ1 (t) + ut|Γ1 (t) |∇ | | | ∇ Γ1 Γ1 t 2 2 + 2 kΩ ut(s) Γ + kΓ α Γut|Γ 1 (s) ds |∇ | 1 ∇ Γ1 Z0

The above energy identity suggests that the energy is decreasing. However, how fast, it needs to be determined. The energy balance also suggests that there is an extra regularity in the damping. How this regularity is propagated onto the entire system is a question we aim to resolve. After establishing a preliminary well-posedness of ﬁnite energy solution (generation of a semigroup of contractions), we shall proceed with a study of:

Additional regularity (analyticity) properties of solutions, both in L2 and Lp • framework. Stability analysis of the resulting semigroups with particular emphasis on • differentiating between strong and uniform stability. The latter will depend on the values of damping parameters. Numerical results illustrating sharpness of the estimates obtained. • The paper is organized into eight sections. In section2, we give an appropriate abstract description of the problem. The results are presented in section3, while the following sections (4-8) each prove one theorem. We show in the theorem 3.1 our model (1) is governed by a C0-semigroup of contractions. Then, we assume, in theorem 3.2 that the semigroup is exponentially stable, in the presence of a structural damping in the interior. Then, theorems 3.3 and 3.4 both show the analyticity of the semigroup under certain conditions for the dampings. We also show that the spectrum is contained in the left-half plane (see theorem 3.5). Finally, section9 presents numerical results obtained with finite element methods, as well as an exhaustive study of the different possible scenarios for this model

2. Deﬁnitions and preliminaries. We ﬁrst introduce a couple of operators from [42] which we will use in this study to pose the abstract problem and prove the theorems. STRONG DAMPED WAVE EQUATION WITH DBC 637

Definition 2.1 (Neumann map). Define the map N by ∆z = 0 on Ω z = Ng ∂ z = g on Γ  ∂ν Γ1 ⇔ z |= 0 on Γ  |Γ0 0 2 3 3 − Then, elliptic theory gives N (L(Γ1),H 2 (Ω) (A 4 )) > 0 ∈ L ⊂ D ∀ Definition 2.2 (The Laplacian in Ω). Let the operator A : L2(Ω) (A) L2(Ω) be defined by: ⊃ D → ∂ Au = ∆u, cD(A) = u L2(Ω), Au L2(Ω), u = 0, u = 0 − { ∈ ∈ |Γ0 ∂ν |Γ1 } Then A is self-adjoint, positive definite, and therefore the fractional powers of A are well-defined. In particular, we have the following caracterization:

1 1 1 (A 2 ) = H (Ω) = z H (Ω), z = 0 on Γ (6) D Γ0 { ∈ 0} 2 2 1 2 1 2 2 2 with z 1 = A z = Ω z = z H1 (Ω) , z (A ) k kD(A 2 ) L2(Ω) |∇ | k k Γ0 ∀ ∈ D where the last equatility follows fromR Poincar´e’sinequality. In order to simplify the 2 2 2 notation, we will denote the L -norm in the following way: u L2(Ω) = u Ω and 2 2 k k | | u 2 = u k kL (Γ1) | |Γ1 1 2 1 Deﬁnition 2.3 (Trace map γ). Let γ : H (Ω) L (Γ ) H 2 (Γ ) be the restric- → 1 ⊂ 1 tion to Γ1: z H1 (Ω), γ(z) = z ∀ ∈ Γ0 |Γ1 Then, by [42, Lemma 2.0]:

∗ 1 N A = γ(z) z (A 2 ) (7) ∀ ∈ D Deﬁnition 2.4 (Laplace-Beltrami on the boundary). For all kΓ 0, set B : L2(Γ ) (B) L2(Γ ) to be: ≥ 1 ⊃ D → 1 Bz = k ∆ z, (B) = z L2(Γ ), k ∆z L2(Γ ) − Γ Γ D { ∈ 1 Γ ∈ 1 } with the associated norm: 2 2 1 2 2 z 1 = B z = kΓ Γz dΓ1 = kΓ u 1 D(B 2 ) 2 H (Γ1) k k L (Γ1) Γ |∇ | k k Z 1

Let (u, ut, u Γ1 , u Γ1 ) = (u1 , u2, u3, u4). | | The third coordinate u3 is deﬁned as the trace of u1: u = u 3 1|Γ1 Deﬁnition 2.5 (Energy spaces). The associated energy space is:

1 2 1 2 = (u , u , u , u ) (A 2 ) L (Ω) (B 2 ) L (Γ ), u = u H { 1 2 3 4 ∈ D × × D × 1 1|Γ1 3} 2 2 2 2 2 with associated norm: u = u1 + u2 + kΓ Γu3 + u4 . k kH |∇ |Ω | |Ω |∇ |Γ1 | |Γ1 1 We note that (B 2 ) is equipped with the graph norm D 2 2 1/2 2 2 2 u 1 u Γ + B u Γ = u Γ + kΓ Γu Γ | |D(B 2 ) ≡ | | 1 | | 1 | | 1 |∇ | 1 638 NICOLAS FOURRIER AND IRENA LASIECKA

With the deﬁnitions of A, B and N, we set: 0 I 0 0 ∆ kΩ∆ 0 0 =   (8) A 0 0 0 I  ∂ k ∂ B αB − ∂ν − Ω ∂ν − −    T 1 1 1 1 ( ) = [u , u , u , u ] (A 2 ) (A 2 ) (B 2 ) (B 2 ), D A { 1 2 3 4 ∈ D × D × D × D such that ∆(u + k u ) L2(Ω), 1 Ω 2 ∈ ∂ 1 1 1 2 (u + k u ) + B 2 (B 2 u + αB 2 u ) L (Γ ), ∂ν 1 Ω 2 3 4 ∈ 1 u = N ∗Au = u , u = N ∗Au = u 1|Γ1 1 3 2|Γ1 2 4} which is densely deﬁned in . H Remark 2. The following representation will be frequently used. ∂ ∆u = A(I N )u − − ∂ν 3. Statement of the main results.

Theorem 3.1. The operator , given in (8), generates a C0-semigroup of contrac- At A tions e t≥0 on . In addition, under the condition (2) , i.e. max kΩ, kΓα > 0, the said{ semigroup} H is strictly contractive. { }

Theorem 3.2. Let kΩ > 0. Consider the operator given in (8) , then the C0- At A semigroup e ≥ is exponentially stable, i.e., { }t 0 C, ω 0 such that eAtu Ce−ωt u ∃ ≥ H ≤ k kH Remark 3. When k = 0 the system is no longer exponentially stable. This has Ω been first discovered by Littman and Markus in [27] for the Scole model [26], the only form of stability retained for finite energy initial data is strong stability (see theorem 3.6). If one considers more regular initial data (with a membership in the domain of the generator) then there may be a possibility of proving algebraic decay rates for the energy function. However these rates, by necessity, can not be uniform with respect to the underlined topology of finite energy space. For this type of results, appplicable to some coupled models with interface conditions , we refer to recent work of Avalos (e,g [4] ) which exploits sharp characterization [9] of algebraic decay rates in terms of the estimates of the resolvent along the imaginary axisis. Whether similar technique could be applied in the present context is not known at the present time.

−1 Theorem 3.3. Let kΩ, kΓ, α > 0. Suppose β 1 so that (iβ ) ( ), then with F = (f , f , f , f )T | | ≥ − A ∈ L H 1 2 3 4 ∈ H C R(iβ; ) k A kL(H) ≤ β | | At It follows that e ≥ is an analytic semigroup on . { }t 0 H Theorem 3.4. Let kΩ, α > 0.

First Part: L2 theory.:

Suppose kΓ 0, and consider kΓ≥0 given by in (8), then kΓ≥0 generates ≥ AA≥ t A A an analytic C -semigroup e kΓ 0 ≥ on . 0 { }t 0 H STRONG DAMPED WAVE EQUATION WITH DBC 639

Second Part: Lp theory.:

Suppose kΓ = 0, and consider kΓ=0 given by in (8), then kΓ=0 generates AA t A A an analytic C -semigroup e kΓ=0 ≥ on 0 { }t 0 1,p p p p = (u1, u2, u3) W (Ω) L (Ω) L (Γ1), u1 Γ = u3 H { ∈ Γ0 × × | 1 } for all p [1, ] ∈ ∞ Theorem 3.5. Suppose that the damping condition (2) holds, i.e. max k , k α > { Ω Γ } 0. With defined in (8), σ( ) iR = . A A ∩ ∅ The first result (theorem 3.1) asserts that problem (1) is well-posed and generates a C0-semigroup of contractions with respect to the finite energy space . Then, AtH theorem 3.2 provides us the exponential stability of the semigroup e ≥ , pro- { }t 0 vided interior damping (kΩ > 0). This result is obtained using energy estimates. In presence of viscoelastic damping both in the interior and on the boundary, theorem 3.3 reveals the C0-semigroup of contractions from theorem 3.1 is also analytic, this result obtained with the resolvent equation and relies on the exponential stability property. Another way to show analyticity - without the exponential stability property - is presented in 3.4 (first part of the theorem) where we proceed to a change of variable transforming the problem into a heat equation with General Wentzell Boundary Conditions (GWBC). Such an approach is related to the work made during the last ten years by Favini and Goldstein’s group: [14] and [12]. This approach also presents the advantage to treat the case kΓ = 0, in which case, we have a stronger result (see second part of the theorem 3.4). We also mention that analyticity should be achieved using energy method and following Haraux’s approach in [21]. Finally, theorem 3.5 gives an important spectral property providing stability results for the model (1). For instance, we will be able to recover the exponential At stability for the analytic semigroups e t≥0 described in theorem 3.4 using the following: { } Theorem A (Theorem 4.4.3 - [37]). Let A be the generator of an analytic semigroup (t). If T sup Reλ : λ σ(A) < 0 { ∈ } then there are constants M 1 and µ > 0 such that (t) Me−µt, i.e. (t) is ≥ kT k≤ T exponentially stable. At This same theorem 3.5 will also give strong stability for the semigroup e t≥0 (for any α > 0 described in theorem 3.1 after application of the following{ classical} result quoted: Theorem B ([2]). Let (t) be a bounded C -semigroup with generator A. Assume T 0 that σr(A) iR = , where σr(.) denotes the residual spectrum. If σ(A) iR is countable, then∩ (t)∅ is strongly stable. This is to say; for every u ∩ T ∈ H (t)u H 0, t ||T || → → ∞ The above theorem will immediately provide the the proof of the following result.

Theorem 3.6. Let kΩ = 0. Then the semigroup is strongly stable. The illustration of strong stability is given on Figure5 where it is shown that one branch of the eigenvalues approaches imaginary axis. While boundary stabilization 640 NICOLAS FOURRIER AND IRENA LASIECKA leads to exponential stability, this is so for models without inertial terms on the boundary. This phenomenon was ﬁrst discovered by Markus and Littman in the context of the Scole model.

4. C0-semigroup of contractions - proof of theorem 3.1. Let kΩ, kΓα 0 , and consider the problem (1): ≥

u k ∆u ∆u = 0 in Ω tt − Ω t − u ∂ (u + k u ) k ∆ (αu + u) = 0 in Γ  tt ∂ν Ω t Γ Γ t 1 u =− 0 − in Γ  0  u(0, x) = u0, ut(0, x) = u1 on Ω  and recall the deﬁnition of (8) using: ∆u = A(I N ∂ )u. − − ∂ν 0 I 0 0 A(I N ∂ ) k A(I N ∂ ) 0 0 = − − ∂ν − Ω − ∂ν  A 0 0 0 I  ∂ k ∂ B αB  − ∂ν − Ω ∂ν − −   

T 1 1 1 1 ( ) = [u , u , u , u ] (A 2 ) (A 2 ) (B 2 ) (B 2 ), D A { 1 2 3 4 ∈ D × D × D × D ∂ such that A(I N )(u + k u ) L2(Ω), − ∂ν 1 Ω 2 ∈ ∂ 1 1 1 2 (u + k u ) + B 2 (B 2 u + αB 2 u ) L (Γ ), ∂ν 1 Ω 2 3 4 ∈ 1 u = N ∗Au = u , u = N ∗Au = u 1|Γ1 1 3 2|Γ1 2 4} which is closed and densely defined in . The following regularity properties ofH the elements in the domain result from the definition of ( ) D A ∂ (u + k u ) [ (B1/2)]0 (9) ∂ν 1 Ω 2 ∈ D this in particular implies ∂ (u + k u ) L (Γ ), k = 0. The above regularity ∂ν 1 Ω 2 ∈ 2 1 Γ result is stronger than classical elliptic theory implies ∂ (u + k u ) H−1/2(Γ ), ∂ν 1 Ω 2 ∈ 1 where the latter is due to the fact that ∆(u1 + kΩu2) L2(Ω) along with u1, u2 H1(Ω). As a consequence of (9) one has well defined duality∈ pairing ∈

∂ < (u + k u ), v >, v D(B1/2), U ( ) (10) ∂ν 1 Ω 2 ∀ ∈ ∀ ∈ D A In order to show the semigroup generation for the dynamics , we wish to use the Lumer-Phillips theorem, [32] and hence we must show the maximalA dissipativity of . StepA 1. is dissipative. A STRONG DAMPED WAVE EQUATION WITH DBC 641

U = (u , u , u , u ) ( ), take the inner product with U: ∀ 1 2 3 4 ∈ D A

< , U >H = (u2, u1) 1 + (u4, u3) 1 AU D(A 2 ) D(B 2 ) ∂ ∂ A(I N )u + k A(I N )u , u − − ∂ν 1 Ω − ∂ν 2 2 Ω recalling (10) ∂ (u + k u ) + Bu + αBu ,N ∗Au − ∂ν 1 Ω 2 3 4 2 Γ1 recalling (7)

1 1 1 1 1 = A 2 u2,A 2 u1 A 2 u1 + kΩA 2 u2,A 2 u2 Ω − Ω ∂ 1 1 + (u1 + kΩu2), u4 + B 2 u4,B 2 u3 ∂ν Γ Γ1 1 D E ∂ 1 1 (u1 + kΩu2), u4 B 2 u3,B 2 u4 − ∂ν − Γ1 Γ1 1 1 D E α B 2 u4,B 2 u4 − Γ1

D 1 2 E 1 2 = kΩ A 2 u2 α B 2 u4 0 − Ω − Γ1 ≤

Therefore, is dissipative. Step 2. RangeA condition. Our aim is to show R(λI ) = To show that is a contraction semigroup, it remains− A toH show that the range A T condition is satisﬁed, i.e., if λ C, Reλ > 0 and f1 f2 f3 f4 is given, then the stationary equation: ∈ ∈ H

u1 f1 u f (λI )  2 =  2 (11) − A u3 f3 u  f   4  4     T is satisﬁed for some u1, u2, u3, u4 ( ) Then (11) becomes: ∈ D A λu u = f 1 − 2 1 λu + A(I N ∂ )u + k A(I N ∂ )u = f  2 − ∂ν 1 Ω − ∂ν 2 2  λu3 u4 = f3  ∂ − λu4 + ∂ν (u1 + kΩu2) + Bu3 + αBu4 = f4   1 1 In the ﬁrst equation, note that f1, u1 (A 2 ), so u2 (A 2 ). Similarly, from the 1 ∈ D ∈ D third equation we have u (B 2 ). 4 ∈ D From the second equation f , u L2(Ω) implies A(I N ∂ )(u +k u ) L2(Ω). 2 2 ∈ − ∂ν 1 Ω 2 ∈ The fourth equation, with u , f L2(Γ ) implies that ∂ (u + k u ) [ (B1/2)]0. 4 4 ∈ 1 ∂ν 1 Ω 2 ∈ D 642 NICOLAS FOURRIER AND IRENA LASIECKA

The above leads to the following representation:

u = λu f 2 1 − 1 u4 = λu3 f3  2 − ∂ ∂  λ u1 + A(I N ∂ν )u1 + λkΩA(I N ∂ν )u1  − ∂ − (12)  = f2 + λf1 + kΩA(I N ∂ν )f1  2 ∂ − λ u3 + ∂ν (u1 + kΩλu1) + Bu3 + λαBu3 ∂  = f4 + λf3 + kΩ ∂ν f1 + αBf3   To solve the stationary problem (12), we shall use a weak formulation and Lax- Milgram theorem. Let (v , v , v , v ) ( ), and for the time beeing we also take 1 2 3 4 ∈ D A F = (f1, f2, f3, f4) in ( ). Later we shall extend the argument by density to all F . We consider theD twoA last equations, multiply them by (v , v ) and integrate ∈ H 1 3 in space over Ω and Γ1, respectively:

2 ∂ ∂ λ u1, v1 Ω + A(I N ∂ν )u1, v1 Ω + kΩλA(I N ∂ν )u1, v1 Ω − ∂ − = f2 + λf1 + kΩλA(I N ∂ν )f1, v1 Ω  2 ∂ −  λ u3, v3 + (u1 + kΩλu1), v3 + (1 + λα) Bu3, v3 Γ  Γ1 ∂ν Γ1 1  ∂ h i  = f4 + λf3 + kΩ f1 + αBf3, v3  ∂ν Γ 1    2 1 1 λ u1, v1 + A 2 u1,A 2 v1 Ω Ω 1 1 2 2 ∂  + kΩλA u1,A v1 ∂ν (u1 + kΩλu1), v3  Ω − Γ1  1 1 ∂  2 2  = (f2 + λf1, v1)Ω + kΩA f1,A v1 kΩ ∂ν f1, v3  Ω − Γ1  2 ∂ 1 1  λ u3, v3 + (u1 + kΩλu1), v3 + (1 + λα) B 2 u 3,B 2 v3  Γ1 ∂ν Γ1  Γ1 ∂ 1 D 1 E = f4 + λf3, v3 + kΩ f1, v3 + αB 2 f3,B 2 v3  Γ1 ∂ν Γ1  h i Γ1  D E   Then combining these two equations, we get:

2 1 1 λ u1, v1 + (1 + λkΩ) A 2 u1,A 2 v1 Ω Ω 2 1 1 + λ u3, v3 + (1 + λα) B 2 u3,B 2 v3 Γ1 Γ1 1 D 1 E 1 = (f + λf , v ) + k A 2 f ,A 2 v + f + λf , v + αB 2 f , v 2 1 1 Ω Ω 1 1 4 3 3 Γ1 3 3 Ω h i Γ1 D E This leads us to consideration of a bilinear form

2 1 1 a(u1, u3, v1, v3) λ u1, v1 + (1 + λkΩ) A 2 u1,A 2 v1 ≡ Ω Ω 2 1 1 + λ u3, v3 + (1 + λα) B 2 u3,B 2 v3 Γ1 Γ1 D E deﬁned for u = (u , u ), v = (v , v ) , where (v , v ) D(A1/2) 1 3 1 3 ∈ V V ≡ { 1 3 ∈ × D(B1/2), v = v .We are solving for the variable u the variational equation: 3 1|Γ1 } a(u, v) = F (v), v D(A1/2) D(B1/2) ∀ ∈ V ≡ × STRONG DAMPED WAVE EQUATION WITH DBC 643 where F (v) be the corresponding right-hand side:

1 1 2 2 F (v) = (f2 + λf1, v1)Ω + kΩA f1,A v1 Ω 1 1/2 + f + λf , v + αB 2 f ,B v 4 3 3 Γ1 3 3 h i Γ1 with F = (f , f , f , f ) D E 1 2 3 4 ∈ H We have continuity of bilinear form on : V × V a(u, v) max λ2 + k λ + α λ , 1 u v | | ≤ { Ω| | | | } k kV k kV F (v) max λ k + α, 1 F v | | ≤ {| | Ω } k kH k kV The bilinear is also coercive:

2 1 1 1 1 Re a(u, u) = λ u1, u1 + A 2 u1,A 2 u1 + kΩλA 2 u1,A 2 u1 Ω Ω Ω 2 1 1 1 1 + λ u3, u3 + B 2 u3,B 2 u3 + λαB 2 u3,B 2 u3 Γ1 Γ1 Γ1 D 2 E D 2 E 2 2 1 2 2 1 C Reλ u + A 2 u + Reλ u + B 2 u 1 Ω 1 3 Γ1 3 ≥ | | Ω | | Γ1 C u 2 ≥ k kV Therefore a(u, v) is both bounded and coercive, so by Lax Milgram for every F ∈ H there exists a unique solution u . Moreover u = (u1, u3) satisﬁes the last two equations in (12). ∈ V Next we reconstruct the remaining part of the vector U . From (12)

u = λu f D(A1/2), u = λu f (B1/2), F (13) 2 1 − 1 ∈ 4 3 − 3 ∈ D ∀ ∈ H ∗ ∗ ∗ Since u3 = N Au1 and f3 = N Af1 we conclude that u4 = N Au2 , as required by the membership in the ( ). The remaining regularity requirements simply follow from the structure of equationsD A in (12). In conclusion, for all F we obtain U = (u1, u2, u3, u4) in ( ) such that (λI )U = F . Thus U∈is H our desired solution. As a consequence,D A generates −A ∈ H At A a strongly continuous semigroup of contraction e t≥0. Note that the result on generation holds without any dissipation active,{ i.e.,} the range condition still holds if kΩ = 0 or kΓα = 0

5. Exponential stability - proof of theorem 3.2. Let kΩ > 0 and recall the model (1):

u k ∆u ∆u = 0 in Ω tt − Ω t − u ∂ (u + k u ) k ∆ (αu + u) = 0 in Γ  tt ∂ν Ω t Γ Γ t 1 u =− 0 − in Γ  0  u(0, x) = u0, ut(0, x) = u1 on Ω  Recall that B= kΓ∆Γ (see deﬁnition 2.4). By [37, Theorem 2.4.1], to show the − At exponential stability of e ≥ it is enough to show that { }t 0 T 2 eAtU ds < H ∞ Z0

644 NICOLAS FOURRIER AND IRENA LASIECKA

Multiply by ut the first equation of (1): (u , u ) + k u 2 + ( u, u ) tt t Ω Ω |∇ t|Ω ∇ ∇ t Ω 2 + utt, ut + kΓα Γut + kΓ ( Γu, Γut) = 0 h iΓ1 |∇ |Γ1 ∇ ∇ Ω (14) 1 d 2 2 2 2 2 2 ut + u + ut + kΓ Γu = kΩ ut kΓα Γut 2 dt | |Ω |∇ |Ω | |Γ1 |∇ |Γ1 − |∇ |Ω − |∇ |Γ1 Let E (t) = u 2 + u 2 , E (t) = u 2 + k u 2 be the potential and kinetic p t Ω t Γ1 k Ω Γ Γ Γ1 energy respectively,| | | then| define the|∇ energy| of this|∇ system| as the summation of the potential and kinetic energies: E(t) = Ep(t) + Ek(t). Using these definitions, we obtain the following energy equality after integrating in time (14): t 2 2 E(0) = E(t) + kΩ ut + kΓα Γut ds (15) |∇ |Ω |∇ |Γ1 Z0 Now multiply by u the first equation of (1) and again integrate in space: (u , u) + k ( u , u) + u 2 tt Ω Ω ∇ t ∇ Ω |∇ |Ω 2 (16) + utt, u + kΓα Γut, Γu + kΓ Γu = 0 h iΓ1 h∇ ∇ iΓ1 |∇ |Ω Integrate in time, and use integration by parts: t 2 2 2 2 ut ut + u + kΓ Γu ds − | |Ω − | |Γ1 |∇ |Ω |∇ |Γ1 Z0 (17) 0 1 d = k u 2 + k α u 2 ds + (u , u) 0 + u , u 0 Ω Ω Γ Γ Γ1 t Ω t t Γ1 t t 2 dt |∇ | |∇ | | h i | Z In (17), we identify the potential energy on the left hand side and we bound the right hand side using the energy at t = 0: t t E (s)ds u 2 + u 2 ds + C E(0) p t Ω t Γ1 0 0 ≤ 0 | | | | Z t Z t (18) 2 2 Ep(s) + Ek(s)ds 2 ut + ut ds + C0E(0) ≤ | |Ω | |Γ1 Z0 Z0 2 2 5.1. Case 1: α = 0 or kΓ = 0. By Poincaré’sinequality that ut Ω C2 ut Ω, then by the trace moment inequality (see [8]) | | ≤ |∇ |

1 1 2 2 2 ut C ut ut | |Γ1 ≤ |∇ |Ω| |Ω (19) C u 2 ≤ |∇ t|Ω Also by Poincar´e’sinequality, the equation (17) becomes: t t E(s)ds C ( u 2 ds + E(0)) (20) ≤ 3 |∇ t|Ω Z0 Z0 where C = max C ,C ,C By the equality (15), we get: 3 { 0 1 2} t E(s)ds CE(0) ≤ Z0

At where C is a constant. Therefore, by [37, Theorem 4.4.1], the semigroup e ≥ { }t 0 is exponentially stable, if α = 0 or kΓ = 0. STRONG DAMPED WAVE EQUATION WITH DBC 645

5.2. Case 2: α > 0. This case is straight forward, indeed: by Poincar´e’sinequality we have: u 2 C u 2 by Poincar´e’sinequality. | t|Ω ≤ 1 |∇ t|Ω 2 2 (21) ut C2 ut by (19) | |Γ1 ≤ |∇ |Ω Therefore, using these two inequalities from above in (16), we obtain: t t 2 2 E(s)ds C3( ut + Γut ds + E(0)) ≤ |∇ |Ω |∇ |Γ1 Z0 Z0 CE(0) by (15) ≤ At Again, by [37, Theorem 4.4.1], the semigroup e t≥0 with α > 0 is exponentially stable. { }

6. Analyticity of the semigroup - proof of theorem 3.3. Assume that kΩ, kΓ, α > 0 and recall that is exponentially stable (see theorem 3.2). A −1 T Let β R so that (iβ ) ( ), then with F = (f1, f2, f3, f4) and pre- ∈ −AT ∈ L H ∈ H image U = (u1, u2, u3, u4) ( ), consider the resolvent equation (iβ )U = F : ∈ D A − A

u1 f1 u f (iβ )  2 =  2 − A u3 f3 u  f   4  4     u1 f1 u f U =  2 = (iβ )−1  2 = (iβ )−1F u3 − A f3 − A u  f   4  4     We want to show β U H C F H. At | | k k ≤ k k Since e ≥ is exponentially stable, then: { }t 0 U C F k kH ≤ k kH

iβu1 u2 = f1 iβu − u = f  3 − 4 3 (22)  iβu2 ∆u1 kΩ∆u2 = f2  − ∂ − iβu4 + ∂ν (u1 + kΩu2) + Bu3 + αBu4 = f4 Multiply the third equation by u and make use of the fourth equation:  2 2 2 iβ u2 + ( u1, u2) + kΩ u2 + iβu4 + Bu3 + αBu4 f4, u2 = (f2, u2) | |Ω ∇ ∇ Ω |∇ |Ω h − iΓ1 Ω (23) Using the ﬁrst equation, observe that: ( u , u ) = ( u , iβ u f ) ∇ 1 ∇ 2 Ω ∇ 1 ∇ 1 − ∇ 1 Ω 1 1 Bu , u = B 2 u ,B 2 u 3 2 Γ1 3 4 h i Γ1 D 1 1 E 1 (24) = B 2 u3, iβB 2 u3 B 2 f3 − Γ1

D 1 2 1 E 1 = iβ B 2 u3 B 2 u3,B 2 f3 − Γ1 − Γ1 D E

646 NICOLAS FOURRIER AND IRENA LASIECKA

Then, combine (23) and (24):

2 2 2 2 2 1 2 1 iβ u + u u B 2 u + k u + α B 2 u 2 Ω 4 Γ1 1 Ω 3 Ω 2 Ω 4 | | | | − |∇ | − Γ1 |∇ | Γ1 (25) 1 1 2 2 = (f2, u2)Ω + (f4, u4)Ω + ( u1, f1)Ω + B u3,B f3 ∇ ∇ Γ1 D E Taking the real part, one gets:

2 2 1 2 kΩ u2 Ω + α B u4 |∇ | Γ1 1 1 2 2 (f2, u2)Ω + (f 4, u4)Ω + ( u1, f1)Ω + B u3,B f3 ≤ | ∇ ∇ Γ1 | (26) C F U D E ≤ k kH k kH C F 2 ≤ k kH Using the ﬁrst two equations in (22) and applying estimate (26):

β u u + f C F 1 Ω 2 Ω 1 Ω kΩ H |∇1 | ≤ |∇ 1 | |∇ |1 ≤ k C k (27) β B 2 u B 2 u + B 2 f F | 3|Γ1 ≤ | 4|Γ1 | 3|Γ1 ≤ α k kH Let K = min k , α , then, with estimate (26), (27) becomes: { Ω }

1 C β u + B 2 u F (28) |∇ 1|Ω | 3|Γ1 ≤ K k kH h i Note that the presence of K on the denominator forces both damping coeﬃcients to be non-zero. Back to (25), multiply by β and take the imaginary part:

2 2 2 2 2 2 1 β u + u β u + B 2 u 2 Ω 4 Γ1 1 Ω 3 | | | | ≤ |∇ | Γ1 (29) h i 1 1 + f βu + f β u + B 2 f βB 2 u + f βu | 2|Ω| 2|Ω |∇ 1|Ω| ∇ 1|Ω | 3|Γ1 | 3|Γ1 | 4|Γ1 | 4|Γ1 Splitting:

2 2 f2 Ω βu2 Ω C f2 Ω + βu2 Ω | | | | ≤ | |2 | | 2 (30) f4 Γ βu4 Γ C f4 + βu4 ( | | 1 | | 1 ≤ | |Γ1 | |Γ1 Using (27) to estimate the ﬁrst terms on the right-hand side of (29):

2 2 2 1 2 β u2 + u4 C( + 1) F (31) | |Ω | |Γ1 ≤ K k kH h i Combine (28) with (31)

At It follows that e ≥ is analytic on , provided k , k , α > 0. { }t 0 H Ω Γ STRONG DAMPED WAVE EQUATION WITH DBC 647

7. Analyticity of Co-semigroup - proof of theorem 3.4. In the following, we

first show (theorem 3.4 - first part) that kΓ≥0 generates an analytic semigroup on , by using the General Wentzell BoundaryA Conditions since this approach offers a moreH complete study of this problem. Indeed, if one adds the additional condition: kΓ = 0, i.e., in the absence of Laplace-Beltrami term on the boundary, the semigroup generated by is analytic not only in H1 (Ω) L2(Ω) (B1/2) L2(Γ ) kΓ=0 Γ0 1 but also in A H ⊂ × ×D ×

1,p p p p = (u1, u2, u3) W (Ω) L (Ω) L (Γ1), u1 Γ = u3 H { ∈ Γ0 × × | 1 } for 1 p (theorem 3.4 - second part). ≤ ≤ ∞

7.1. Proof of theorem 3.4 - ﬁrst part. Consider (1) with kΩ, α > 0: u k ∆u ∆u = 0 in Ω tt − Ω t − u ∂ (u + k u ) k ∆ (αu + u) = 0 in Γ  tt ∂ν Ω t Γ Γ t 1 u =− 0 − in Γ  0  u(0, x) = u0, ut(0, x) = u1 on Ω with the associated Cauchy problem operator given in (8).  kΓ≥0 The above model contains two possible scenarios:A Scenario 1: kΩ > 0, kΓ = 0 which corresponds to the absence of Laplace Bel- trami operator. In that case, the result has been known, however our more general proof will also yield the expected conclusion. Scenario 2. kΩ, kΓ, α > 0 corresponds to the presence of Laplace Beltrami operator with additional strong damping on the boundary α > 0. It an alternate proof to the previous theorem. Note that there is a third related scenario: Scenario 3: kΩ > 0, kΓ > 0, α = 0 corresponds to the presence of Laplace Beltrami operator without the additional strong damping on the boundary α = 0. In this case, the analyticity does not hold but we expect some smoothing behavior in line with Gevrey class behavior. See ﬁgure6 where the eigenvalues are aligned along the parabola Imλ = a b[Reλ]2. Our proof is based on connecting− the problem under consideration to Wentzell semigroups studied in [12, 14]. To this end we introduce the new variable:

z = u + kΩut on Ω and Γ1

We recall from the deﬁnition 2.4 that B = kΓ∆, then the two ﬁrst equations of (1) are equivalent to: −

u + u = z in Ω t kΩ kΩ u z γ(ut) + γ( ) = γ( ) in Γ1  kΩ kΩ zt z−u (32)  k = ∆z + k2 in Ω  Ω Ω  ∂ kΓα α with ∆z + z ∆Γz kΓ(1 )∆Γu = 0 in Γ1 ∂ν − kΩ − − kΩ Note that the ﬁrst two equations are simply an ordinary diﬀerential equation.  By the initial condition of (1), we are looking for U (u, u ) X = (u , u ) H1 (Ω) (B1/2), γ(u) = u ≡ |Γ1 ∈ u { |Ω |Γ1 ∈ Γ0 × D |Γ1 } with the associated norm: 2 2 2 1 U = u + B 2 u Xu Ω Γ1 k k |∇ | | Γ1

648 NICOLAS FOURRIER AND IRENA LASIECKA

1 1/2 Thus Xu is a closed subspace of H (Ω) (B ). Consider the abstract ODE × D 1 Vt + V = 0,V (0) Xu (33) kΩ ∈ and denote by (t) the governing semigroup. This semigroup is obviously analytic T − 1 on X and given by (t)V = e kΩt V. u T

(t)U = (t)u 1 + (t)u 1/2 0 Xu 0 Ω H (Ω) 0 Γ1 D(B ) kT k kT | k Γ0 kT | k − 1 t kΩ e u0 Ω H1 (Ω) + u0 Γ1 D(B1/2) ≤ k | k Γ0 k | k 1 − k t h i e Ω U0 ≤ k kXu Since V (t) C V , then (t) is an analytic C -semigroup on X . t Xu 0 Xu 0 u Note thatk thek solution≤ k tok the firstT two equations of 32 can be written as a perturbation 1 1 Ut + U = Z,U(0) = U0 kΩ kΩ where Z (z , γ(z)). Thus, by the variation of parameter formula we obtain: ≡ |Ω t U(t) = (t)U + T (t s)Z(s)ds (34) T 0 − Z0 The caveat is that the above equation is considered on Xu which means that with z L2(Ω) one obtains unbounded forcing in ((34)). However, the analyticity of the semigroup∈ will help in handling this part. Before we move to the other two equations, we need to do some preliminary work on the z-dynamics. ¯ Definition 7.1 (Xz). Identify every z C(Ω) with Z = (z Ω, z Γ1 ) and define Xz to be the completion of C(Ω)¯ in the norm:∈ | | 2 2 1/2 z 2 := ( z + z ) |k |k | |Ω | |Γ1 In general, a member of X is H = (f, g), where f Lp(Ω), g Lp(Γ ). Note that z ∈ ∈ 1 f may not have a trace on Γ1, and even if f does, this trace needs not equal g. Also, define the formal Laplacian operator A with General Wentzell Boundary Conditions (GWBC) by: N

Au = ∂i(aij(x)∂ju) in Ω (35) i,j=1 X ∂ Az + β z + γz ζβ∆ z = 0 on δΩ (36) ∂ν − Γ In [12, Theorem 3.1] and then in [14, Theorem 3.2], Favini has shown that the heat equation with GWBC was governed by an analytic semigroup on Xz.

Theorem C (Theorem 3.2 - [14]). The realization A2 of A in Xz with domain (A ) = Z = (z , z ) C2(Ω)¯ , z satisﬁes (36) D 2 { |Ω |δΩ ∈ |δΩ } is symmetric and bounded above on Xz. Also, the closure G2 of A2 is selfadjoint and generates a cosine function and G2t a quasicontraction (contraction, if γ 0) semigroup e t≥0 on X2 which is analytic in the right half plane. ≥ { } STRONG DAMPED WAVE EQUATION WITH DBC 649

Set β = kΩ, γ = 0, ζ = kΓα and rewrite (32) using the operator A = kΩ∆: u + u = z in Ω t kΩ kΩ γ(u ) + γ( u ) = γ( z ) in Γ t kΩ kΩ 1  z u (37) zt = Az + in Ω  kΩ − kΩ  with Az + k ∂ z k α∆ z = k (k α)∆ u in Γ Ω ∂ν − Γ Γ Γ Ω − Γ 1  Observe that the last equation in (37) corresponds to the heat equation (35) with z u General Wentzell Boundary Conditions (36) perturbed on Ω by and on Γ1 kΩ − kΩ by k (k α)∆ u . − Γ Ω − Γ |Γ1 Remark 4. Note that the fourth equation can be rewritten as an evolution equation in γ(z) : ∂ z u zt = kΩ z + kΓα∆Γz + + kΓ(kΩ α)∆Γu (38) − ∂ν kΩ − kΩ − This way of writing makes a connection with evolution equations governed by dynamic boundary conditions. To show the analyticity of (37) on = X X , we will proceed in two steps: Hu,z u × z ﬁrst, we show that the system without perturbation on Γ1 is analytic (lemma 7.2). Then, we will show that the analyticity with the perturbation on Γ1 is preserved (Lemma 7.3) The system given in (37) can be written in a compact form as U + 1 U = 1 Z t kΩ kΩ 1 Zt + G2Z = (Z U) + P (γ(u)) kΩ − T where P (γ(u)) [0, kΓ(kΩ α)∆Γγ(u)] We begin the analysis with unperturbed system. We shall≡ show that− the associated semigroup inherits analyticity properties from Wentzell semigroup. Lemma 7.2. Given the system: u + u = z in Ω t kΩ kΩ γ(u ) + γ( u ) = γ( z ) in Γ t kΩ kΩ 1  z u (39) zt = Az + in Ω  kΩ − kΩ  with Az + k ∂ z k α∆ z = 0 in Γ Ω ∂ν − Γ Γ 1  1 where (u Ω, u , z , z )T H1 (Ω) (B1/2) (G 2 ). Then the following Γ1 Ω Γ1 Γ0 2 inequality| holds,| | | ∈ × D × D

C1 Ut(t) + Zt(t) C0 U0 + Z0 (40) k kXu k kXz ≤ k kXu t k kXz It follows that the semigroup associated generates an analytic C -semigroup on . 0 Hu,z Note that lemma 7.2 corresponds to the case α = kΩ.

Proof. First, recall the action of G2 the closure of A2 the realizaion of A in Xz, for 1 some Z = (z , z ) (G 2 ): |Ω |Γ1 ∈ D 2 ∂ (G2Z,Z) = kΩ (∆z, z) kΩ z, z + kΓ ∆Γz, z Xz Ω − ∂ν h iΓ1 Γ1 2 2 = kΩ z kΓα Γz 0 − |∇ |Ω − |∇ |Γ1 ≤ 650 NICOLAS FOURRIER AND IRENA LASIECKA

Note that G2 is dissipative, and now observe that by Favini’s result quoted in theoremC about analyticity:

θ 1 G2Z(t) Z0 X for θ [0, 1] Xz ≤ tθ k k z ∈

Therefore we have the following inequality for all t > 0 and taking into account 1 that Z(t) (G 2 ): ∈ D 2 1 2 C Z(t) 2 = z(t) 2 + k z(t) 2 C G 2 Z(t) Z 2 (41) Xu Ω Γ Γ Γ1 2 0 Xz k k |∇ | |∇ | ≤ Xz ≤ t k k

By the variation of parameters, the ﬁrst two equations of (39) have a solution in Xu: t − 1 t − 1 (t−s) U(t) e kΩ U + e kΩ Z(s) ds Xu 0 Xu Xu k k ≤ k k 0 k k Zt − 1 (t−s) 1 C U + C e kΩ G 2 Z(s) ds 0 Xu 2 ≤ k k Xz 0 (42) Z t 1 1 − k (t−s) C U0 + C e Ω Z0 ds ≤ k kXu √s k kXz Z0 1 1 − k (t−s) 1 C U0 + Z0 since e Ω L (0,T ) ≤ k kXu k kXz √s ∈ Finally, the desired estimate is obtained by combining (41) and (42): 1 U (t) + Z (t) Z(t) U(t) + Z(t) U(t) + G Z(t) t Xu t Xz Xu Xz 2 Xz k k k k ≤ kΩ k − k k − k k k

C U(t) + Z(t) + G2Z(t) ≤ k kXu k kXu k kXz 1 1 C U0 + (1 + ) Z0 + Z0 ≤ k kXu √t k kXz t k kXz which completes the proof of equation (40), implying analyticity of the corresponding semigroup.

Let S be the generator of the analytic semigroup governing model (39) on the space = X X with the domain: Hu,z u × z 1 T 1 1 (S) = V = (u , u , z , z ) H (Ω) (B 2 ) (G 2 ) D { |Ω |Γ1 |Ω |Γ1 ∈ Γ0 × D × D 2 } Then the orginal system (37) with the fourth equation replaced by (38) is equivalent to:

Vt = (S + P )V (43) where 1 G + I 0 JJ 1 0 2 kΩ S = − with J = ,K = ∂ I JK kΩ 0 1 kΩ kΓα∆Γ + − − ∂ν kΩ 0 0 0 0 P = 2×2 2×2 with L = L 0 × 0 k (k α)∆ 2 2 Γ Ω − Γ Lemma 7.3. The semigroup generated by S + P is analytic on . Hu,z STRONG DAMPED WAVE EQUATION WITH DBC 651

1 1/2 Proof. Let V = (U, Z) = (u Ω, u Γ , z Ω, z Γ ) H (Ω) (B ) Xz, with | | 1 | | 1 ∈ Γ0 × D × V = (u , u , z , z ) be solution of (43), then 0 0|Ω 0|Γ1 0|Ω 0|Γ1 t St S(t−τ) V (t) = e V0 + e PV (τ)dτ Z0 Taking the Laplace Transform

V (λ) = R(λ, S)V0 + R(λ, S)PV (λ) leads to the following relation:

[I R(λ, S)P ] V (λ) = R(λ, S)V − 0

Thus, S + P generates an analytic C0-semigroup if and only if [I R(λ, S)P ] is invertible. This last statement follows from − V (λ) = [I R(λ, S)P ]−1R(λ, S)V (0) − and the estimate R(λ, S) C λ −1 | |L(Hu,z ) ≤ | | Thus it is enough to check R(λ, S)PV 1 : k kHu,z ≤ 2 1 − 1 R(λ, S)PV = R(λ, S)S 2 S 2 PV Hu,z k k Hu,z 1 1 2 − 2 S R(λ, S) S PV ≤ L(Hu,z ) Hu,z

1 − 1 S 2 PV ≤ √λ Hu,z

since S generates an analytic semigroup on u,z . It remains to bound the second term. H 1 1 1 T 1 2 For all φ = (φ , φ , φ , φ ) (S 2 ) H (Ω) (B 2 ) (G ), 1 2 3 4 Γ0 2 ∈1 D ⊂ 1 × D × D 1 − ∗ 1 1/2 2 deﬁne ϕ = (ϕ , ϕ , ϕ , ϕ ) = [S 2 ] φ (S 2 ) H (Ω) (B ) (G ), thus 1 2 3 4 Γ0 2 1 ∈ D ⊂ × D × D S 2 ϕ C φ . Then, we have: Hu,z Hu,z ≤ k k

− 1 S 2 P V, φ = (P V, ϕ ) 4 Hu,z Hu,z (kΓ(kΩ α) Γu Γ , Γϕ4) 2 ≤ − ∇ | 1 ∇ L (Γ1) C u Γ 1 ϕ4 1 ≤ k | 1 kH (Γ1) k kH (Γ1) C U φ ≤ k kHu,z k kHu,z 1/2 1 where we have assumed that (B ) H (Γ1) (since otehrwise kΓ = 0 ), by choosing λ > R2 with R suﬃcientlyD large:∼ C 1 R(λ, S)PV V [I R(λ, S)P ] k kHu,z ≤ R k kHu,z ⇒ k − kL(Hu,z ) ≥ 2

A ≥ t Then generates an analytic semigroup e kΓ 0 ≥ on , for k , α > 0. A { }t 0 H Ω 652 NICOLAS FOURRIER AND IRENA LASIECKA

7.2. Proof of theorem 3.4 - second part. Consider (1) with kΩ > 0, kΓ = 0: u k ∆u ∆u = 0 in Ω tt − Ω t − u ∂ (u + k u ) k ∆ (αu + u) = 0 in Γ  tt ∂ν Ω t Γ Γ t 1 u =− 0 − in Γ  0  u(0, x) = u0, ut(0, x) = u1 on Ω To prove that generates an analytic semigroup on , we proceed as in the  kΓ=0 p previous proof.A It is enough to consider the first two equationsH of (32), which are ordinary differential equations can be defined in X = u W 1,p(Ω), γ(u) = u , u Γ0 Γ1 for 1 p . Then trace theory provides the following{ estimate∈ for u : | } ≤ ≤ ∞ Γ1 1,p u Γ 1−(1/p),p C u 1,p , u W (Ω) k | 1 kW (Γ1) ≤ k kW (Ω) ∀ ∈ For the z-dynamics, we first generalize the space Xz by Xp: ¯ Definition 7.4 (Xp). Identify every z C(Ω) with z = (z Ω, z Γ1 ) and redefine Xp to be the completion of C(Ω)¯ in the norm:∈ | |

1 p z := u p dx + u p dS |k |kp | |Ω| | |Γ1 | ZΩ ZΓ1 for 1 p . ≤ ≤ ∞ Then, an analogy of theoremC was established, which we quote: Theorem D (Theorem3.3 - [14]). In the absence of Laplace-Beltrami term on the boundary, the closure Gp of the realization Ap of A in Xp with domain (A ) = z = (z , z ) (A ) X z satisﬁes (36) D p { |Ω |δΩ ∈ D 2 ∩ p |δΩ } is analytic on X for p [1, ]. p ∈ ∞ 1 1,p 1− 1 ,p By the above theorem, the domain (G 2 ) is contained in W (Ω) W p (Γ ) p Γ0 1 and we have the following relationship,D using interpolation theorem: × p 1 p p p 2 1 Z(t) = z(t) 1,p G Z(t) Z Xu W (Ω) p 0 Xp k k k k ≤ Xp ≤ √t k k

Following the proof of lemma 7.2, we use the variation of parameters and proceed as in (42) t − 1 t − 1 (t−s) U(t) e kΩ U + e kΩ Z(s) ds Xu 0 Xu Xu k k ≤ k k 0 k k Zt − 1 (t−s) 1 C U + C e kΩ G 2 Z(s) ds 0 Xu p ≤ k k 0 Xp Z

C U0 + Z0 ≤ k kXu k kXp We deduce: h i 1 U (t) + Z (t) Z(t) U(t) + Z(t) U(t) + G Z(t) t Xu t Xp Xu Xp p X k k k k ≤ kΩ k − k k − k k k p h 1 1 i C U0 + (1 + ) Z0 + Z0 ≤ k kXu √t k kXp t k kXp It follows that the semigroup generated by is analytic on . AkΓ=0 Hp STRONG DAMPED WAVE EQUATION WITH DBC 653

8. Spectral property - proof of the theorem 3.5. Suppose that the condition (2) holds, i.e. max kΩ, kΓα > 0. In order to show that σ( ) iR = , we shall follow the general line{ of arguments} presented in [3]. To wit, weA begin∩ by∅ computing the adjoint of : A

Lemma 8.1 (Adjoint of ). With as deﬁned in (8) and using the notation A A ∆u = A(I N ∂ )u, the adjoint ∗ is given to be − − ∂ν A

0 I 0 0 − A(I N ∂ ) k A(I N ∂ ) 0 0 ∗ =  − ∂ν − Ω − ∂ν  (44) A 0 0 0 I −  ∂ k ∂ B αB  ∂ν − Ω ∂ν −   

∗ T 1 1 1 1 ( ) = [u , u , u , u ] (A 2 ) (A 2 ) (B 2 ) (B 2 ), D A { 1 2 3 4 ∈ D × D × D × D ∂ such that A(I N )(u k u ) L2(Ω), − ∂ν 1 − Ω 2 ∈ ∂ 1 1 1 2 (u k u ) + B 2 (B 2 u αB 2 u ) L (Γ ), ∂ν 1 − Ω 2 3 − 4 ∈ 1 u = N ∗Au = u and u = N ∗Au = u 1|Γ1 1 3 2|Γ1 2 4}

Proof. Let

T 1 1 1 1 S = [u , u , u , u ] (A 2 ) (A 2 ) (B 2 ) (B 2 ), { 1 2 3 4 ∈ D × D × D × D ∂ such that A(I N )(u k u ) L2(Ω), − ∂ν 1 − Ω 2 ∈ ∂ 1 1 1 2 (u k u ) + B 2 (B 2 u αB 2 u ) L (Γ ), ∂ν 1 − Ω 2 3 − 4 ∈ 1 u = N ∗Au = u and u = N ∗Au = u 1|Γ1 1 3 2|Γ1 2 4}

Then u ( ), y = [y , y , y , y ]T S: ∀ ∈ D A 1 2 3 4 ∈

1 1 ( u, y) 2 2 = A H∼D(A 2 )×L (Ω)×D(B 2 )×L (Γ1) 1 1 ∂ A 2 u2,A 2 y1 A(u1 N u1), y2 2 1 1 L (Ω) − − ∂ν [D(A 2 )]∗×D(A 2 ) ∂ 1 1 kΩ A(u2 N u2), y2 + α B 2 u4,B 2 y3 1 1 2 − − ∂ν [D(A 2 )]∗×D(A 2 ) L (Γ1) ∂ ∂ u1, y4 kΩ u2, y4 − ∂ν 2 − ∂ν 2 L (Γ1) L (Γ1) (Bu3, y4) 1 1 α (Bu4, y4) 1 1 − [D(B 2 )]∗×D(B 2 ) − [D(B 2 )]∗×D(B 2 ) 654 NICOLAS FOURRIER AND IRENA LASIECKA ∂ ∂ = u2,A(y1 N y1) + u4, y1 1 1 − ∂ν ∗ ∂ν 2 D(A 2 )×[D(A 2 )] L (Γ1) 1 1 ∂ ∗ A 2 u1,A 2 y2 + u1,N Ay2 2 − L (Ω) ∂ν L2(Γ ) 1 1 1 ∂ ∗ kΩ A 2 u2,A 2 y2 + kΩ u2,N Ay4 2 − L (Ω) ∂ν L2(Γ ) 1 ∂ ∂ + (u4, By3) 1 1 u1, y4 kΩ u2, y4 D(B 2 )×[D(B 2 )]∗ − ∂ν 2 − ∂ν 2 L (Γ1) L (Γ1) 1 1 B 2 u3,B 2 y4 α (u4, By4) 1 1 2 [D(B 2 )]∗×D(B 2 ) − L (Γ1) − ∂ ∂ = u2,A(y1 N y1) + u4, y1 1 1 − ∂ν ∗ ∂ν 2 D(A 2 )×[D(A 2 )] L (Γ1) 1 1 ∂ A 2 u1,A 2 y2 kΩ u2,A(y2 N y2) 2 1 1 − L (Ω) − − ∂ν D(A 2 )×[D(A 2 )]∗ ∂ kΩ u2, y2 + (u4, By3) 1 1 D(B 2 )×[D(B 2 )]∗ − ∂ν 2 L (Γ1) 1 1 B 2 u3,B 2 y4 α (u4, By4) 1 1 2 D(B 2 )×[D(B 2 )]∗ − L (Γ1) − = (u, Λy)H 0 I 0 0 − A(I N ∂ ) k A(I N ∂ ) 0 0 where Λ = ∂ν Ω ∂ν  −0− 0− 0 I  −  ∂ k ∂ B B  ∂ν − Ω ∂ν −  Therefore, S ( ∗) and ∗ = Λ  ⊆ D A A |S Suppose that λ C with Reλ > 0, to show the opposite containment, it is enough ∈ T T ∗ to verify that F = (f1, f2, f3, f4) , there exists Y = (y1, y2, y3, y4) ( ) such that ∀ ∈ H ∈ D A λY ∗Y = F − A ∈ H

λy1 + y2 = f1 λy A(I N ∂ )y k A(I N ∂ )y = f  2 ∂ν 1 Ω ∂ν 2 2 λy +− y =−f − −  3 4 3  λy ∂ y k ∂ y + By αBy = f 4 − ∂ν 1 − Ω ∂ν 2 3 − 4 4   y2 = f1 λy1 y = f − λy 4 3 − 3  λ2y A(I N ∂ )y λk A(I N ∂ )y  − 1 − − ∂ν 1 − Ω − ∂ν 1 (45)  = f λ + k A(I N ∂ ) f  2 − Ω − ∂ν 1  λ2y ∂ y λk ∂ y By αBy − 3 − ∂ν 1 − Ω ∂ν 1 − 3 − 3 = f λf k ∂ f αBf  4 3 Ω ∂ν 1 3  − − − To solve the stationnary problem (45), we shall use a weak formulation and Lax-  ∗ Milgram theorem. To this end, introduce (y1, y2, y3, y4) ( ) and (y1, y2, y3, y4) . Later we shall extend the argument by density to all∈ DF A . We consider the ∈ H ∈ H two last equations of (45), multiply them by (v1, v3) and integrate in space over Ω and Γ1, respectively: STRONG DAMPED WAVE EQUATION WITH DBC 655

∂ λ2y , v (1 + λk ) A(I N )y , v − 1 1 Ω − Ω − ∂ν 1 1 Ω 2 ∂ λ y3, v3 (1 + λkΩ) y1, v3 (1 + λα) By3, v3 − Γ1 − ∂ν − h iΓ1 Γ1 ∂ = f λf k A(I N )f , v 2 − 1 − Ω − ∂ν 1 1 Ω ∂ + f λf k f αBf , v 4 − 3 − Ω ∂ν 1 − 3 3 Γ1 Rewriting, we obtain:

2 1 1 λ y1, v1 (1 + λkΩ) A 2 y1,A 2 v1 − Ω − Ω 2 1 1 λ y3, v3 (1 + λα) B 2 y3,B 2 v3 Γ1 − − Γ1 D 1 1 E 2 2 = (f2 λf1, v1)Ω kΩ A f1,A v1 − − Ω 1 1 + f λf , v α B 2 f ,B 2 v 4 3 3 Γ1 3 3 h − i − Γ1 This leads us to consideration of a bilinear formD E 2 1 1 a(y1, y3, v1, v3) λ y1, v1 (1 + λkΩ) A 2 y1,A 2 v1 ≡ − Ω − Ω 2 1 1 λ y3, v3 (1 + λα) B 2 y3,B 2 v3 Γ1 − − Γ1 D1/2 1/E2 where y = (y1, y3), v = (v1, v3) (v1, v3) D(A ) D(B ), v3 = v1 Γ1 . We are solving for the variable∈ Vu ≡the { variational∈ equation:× | } a(y, v) = F (v), v D(A1/2) D(B1/2) ∀ ∈ V ≡ × where F (v) be the corresponding right-hand side:

1 1 2 2 F (v) = (f2 λf1, v1)Ω kΩ A f1,A v1 − − Ω 1 1 + f λf , v α B 2 f ,B 2 v 4 3 3 Γ1 3 3 h − i − Γ1 with F = (f , f , f , f D) E 1 2 3 4 ∈ H We have continuity of bilinear form on : V × V a(y, v) C y V v V | | ≤ k k k k (46) F (V ) C F v | | ≤ k kH k kV The bilinear form is coercive: 2 2 2 2 1 2 2 1 Re a(u, u) = λ y + (1 + λk ) A 2 y + λ y + (1 + λα) B 2 y 1 Ω Ω 1 3 Γ1 3 | | Ω | | Γ1

C y 2 ≥ k kH By the Lax-Milgram theorem, for every F there exists a unique solution y . ∈ H ∈ V Proceeding as in the range condition from the proof of the generation of the C0- T ∗ semigroup (theorem 3.1): for all F , we obtain Y = (y1, y2, y3, y4) in ( ) such that λY ∗Y = F . Thus∈ HY is our desired solution and we concludeD A that Y S. Therefore,− A ( ∈∗) H = S and ∗ = Γ. ∈ D A A 656 NICOLAS FOURRIER AND IRENA LASIECKA

Now consider the spectrum of : σ( ) = σ ( ) σ ( ) σ ( ) where σ ( ), A A p A ∪ r A ∪ c A p A σr( ) and σc( ) denotes respectively the point spectrum, the residual spectrum andA the continuousA spectrum of ; and show that it does not contain the imaginary A axis: iR. Step 1. σc( ) iR = A ∩ ∅ Let’s ﬁrst cite a theorem [15, Problem 2.54 p. 128]: If λ σc( ), then λ does not have a closed range. ∈ A A − With as given in (8), assume that λ = ir σc( ) with r = 0. f = [Af , f , f , f ]T , suppose that u =∈ [u1, uA2, u3, u4] 6 ( ) such that: ∀ 1 2 3 4 ∈ H ∈ D A

u1 f1 u f (ir )  2 =  2 − A u3 f3 u  f   4  4 which is equivalent to:     iru u = f 1 − 2 1 iru3 u4 = f3  − ∂ ∂  iru2 + A(u1 N ∂ν u1) + kΩA(u2 N ∂ν u2) = f2  ∂ − ∂ − iru4 + ∂ν u1 + kΩ ∂ν u2 + Bu3 + αBu4 = f4  ∗ The two last equations can be rewritten, using the ﬁrst two and u3 = N Au1 as follows: r2u + (1 + k ir)Au AN( ∂ u + k ∂ u ) = irf + k Af + f − 1 Ω 1 − ∂ν 1 Ω ∂ν 2 1 Ω 1 2 ∂ u + k ∂ u = (47)  ∂ν 1 Ω ∂ν 2 f + r2N ∗Au + irf BN ∗Au αirBN ∗Au + αBN ∗Af  4 1 3 − 1 − 1 1 Substitute the second equation into the ﬁrst one. 2 2 ∗ ∗ ∗ r u1 + (1 + kΩir)Au1 AN(r N Au1 BN Au1 αirBN Au1) − − − − (48) = irf1 + kΩAf1 + f2 + AN(f4 + irf3 + αBf3)

1 1 Let V = (A 2 ) (B 2 ) with associated norm: D ∩ D 1 2 1 2 2 2 u V = A u + B u (49) k k Ω Γ1

Consider the left hand side of (48) and define the operators T,M,K : V V ∗, by → T = M + K M = A + ANBN ∗A + ir(k A + αANBN ∗A)  Ω K = r2(I + ANN ∗A)  − 1 2 1 Also, using the right hand side of (48), define F : (A 2 ) L (Ω) (B 2 ) L2(Γ ) V ∗ by: H ⊂ D × × D × 1 → F = (ir + kΩA) I AN(ir + αB) AN Observe that F is well-defined: let v V ∗. First, the term αANBf : ∈ 3 1 ∗ 1 (αANBf3, v) 1 1 = α B 2 f3,N AB 2 v D 2 ∗×D 2 2 [ (A )] (A ) L (Γ1)

1 2 1 2 2 2 α B f3 B v|Γ1 ≤ Γ1 Γ1 1 1 ∗ Since f (B 2 ), then αANBf [ (A 2 )] . 3 ∈ D 3 ∈ D STRONG DAMPED WAVE EQUATION WITH DBC 657

Therefore, we obtain:

F (V ) C F v ∗ | | ≤ k kH k kV It remains to show that T is invertible, we will proceed in three steps. Step 1-a. Compactness of K ∗ 1 By deﬁnition of A, N and N A, K is compact from (A 2 ) into its dual. Step 1-b. M is boundedly invertible. D Given M, u V and v V ∗, deﬁne its bilinear form M(., .), by: ∈ ∈ 1 1 1 ∗ 1 ∗ M(u, v) = (1 + kΩir) A 2 u, A 2 v + (1 + αir) B 2 N Au, B 2 N Av Ω Γ1 D E Then M(., .) is a coercive and bounded bilinear form, setting KM = max 1, kΩr + αr : { } 2 2 2 2 1 1 1 ∗ 1 ∗ M(u, v) KM A 2 u A 2 v + B 2 N Au B 2 N Av | | ≤ Ω Ω Γ1 Γ1 C u v by ( 49) ≤ k k V k k V 2 2 2 2 1 1 ∗ 1 1 ∗ Re M(u, u) = A 2 u + B 2 N Au + kΩr A 2 u + αr B 2 N Au | | Ω Γ1 Ω Γ1

1 2 1 2 2 2 C ( A u + B uΓ1 ) = C u V ≥ Ω Γ1 k k

Therefore by Lax-Milgram, the operator M is boundedly invertible. By the Fred- holm’s alternative, we deduce the desired invertibility of T provided that T is injective. Step 1-c. T is injective. Suppose that T u1 = 0, assume that r = 0 and take the duality pairing with respect to v = u V ∗, then: 6 1 ∈ 2 ∗ ((1 + kΩir)Au1, v) 1 1 r (I + ANN Au1), v 1 1 [D(A 2 )]∗×D(A 2 ) − [D(A 2 )]∗×D(A 2 ) ∗ + ((1 + αir)ANBN Au1, v) 1 1 [D(A 2 )]∗×D(A 2 )

2 2 2 1 2 ∗ 2 1 ∗ = (1 r ) A 2 u r N Au + B 2 N Au 1 1 Γ1 1 − Ω − | | Γ1 2 2 (50) 1 1 ∗ + ir kΩ A 2 u1 + α B 2 N Au1 = 0 Ω Γ1 Taking the imaginary part in (50 ) implies u = 0. Thus T is injective. Note that 1 the injectivity does not necessarly hold if we do not impose the damping condition. Thus, T is invertible which achieves the proof of the ﬁrst step: σc( ) iR = . A ∩ ∅ Step 2. σp( ) iR = A ∩ ∅ With as given in (8), if for r R and r = 0, there exists u = [u1, u2, u3, u4]T ( ) suchA that: ∈ 6 ∈ D A

u1 u1 u u  2 = ir  2 A u3 u3 u  u   4  4     658 NICOLAS FOURRIER AND IRENA LASIECKA which is equivalent to:

u2 = iru1 u4 = iru3  ∂ ∂ (51)  iru2 + A(u1 N ∂ν u1) + kΩA(u2 N ∂ν u2) = 0  ∂ − ∂ − iru4 + ∂ν u1 + kΩ ∂ν u2 + Bu3 + qBu4 = 0  Observe that (51) is equivalent to T u1 = 0. Then, in step 1, we have already shown that u1 = 0 which achieves the proof for step 2. Step 3. σr( ) iR = A ∩ ∅ Again, let’s cite a theorem [15, p. 127]: If the eigenvalue λ C is in the residual spectrum of , then λ σ ( ∗) ∈ A ∈ p A With ∗ as given in (44), if for r R and r = 0, there exists u = [u1, u2, u3, u4]T ( ∗)A such that: ∈ 6 ∈ D A u1 u1 u u ∗  2 = ir  2 A u3 u3 u  u   4  4 which is equivalent to:    

u2 = iru1 −u = iru  − 4 3 (52) Au AN ∂ u k Au + k AN ∂ u = iru  1 − ∂ν 1 − Ω 2 Ω ∂ν 2 2  ∂ u k ∂ u + Bu αBu = iru ∂ν 1 − Ω ∂ν 2 3 − 4 4  Proceeding as in (47) and (48), then (52) can also be written as T u1 = 0, which again implies that u1 = 0, by step 1. Therefore, the residual does not intersect the imaginary axis.

At Remark 5. Since iR σ( ), the semigroup e t≥0 generated in theorem 3.1 is strongly stable by the stability6⊂ A theorem from Arendt-Batty{ } theorem [2, Theorem 2.4] cited in theoremB. With the additional assumption that kΩ, α > 0, the semigroup Ak ≥0t e Γ t≥0 is analytic by theorem 3.4 in , therefore it is exponentially stable by the{ stability} theoremA cite from [37, TheoremH 4.4.3].

9. Numerical results & study of the different cases. In parallel, this model was studied numerically. It was developped using finite elements in order to get a better intuition of the overall dynamics by observing the solution’s behavior and to analyze its spectrum, thus some stability and regularity properties described before. Let Ω = (0, 10) (0, 10) be filled with fluid which is at rest except for acoustic wave motion. Assume× that the sides are not rigid but subject to small oscillations, which defines the dynamic boundary Γ1 (like a container of the rectangular shape sealed on the four sides by a resilient material). Assume that a small rectangle inside the domain Ω is clamped: Γ0 = (3, 4) (3, 4), then create a mesh of this domain. To illustrate this set-up, we present the× two pictures (Figure1-2 ) which correspond to the approximate solutions of model (1)

with internal damping only and boundary damping (kΩ, kΓ > 0, α = 0 - • Figure1 ) with internal and boundary damping (k , k , α > 0 - Figure2 ) • Ω Γ at the same time t > t0 given the same initial conditions u0 and u1. We ﬁrst observe the presence of the diﬀerent waves of the system. The main wave moving in the STRONG DAMPED WAVE EQUATION WITH DBC 659 interior is feeding waves on each edge of the exterior boundary. We also note the Dirichlet boundary condition where the boundary Γ0 is represented by two squares on the left of the highest point. The presence of boundary damping add some smoothness to the solution, not only on the boundary but also in the interior.

Figure 1. Approximate solution of (1) with α = 0

Figure 2. Approximate solution of (1) with α = 1

First, setting u(x, t) = ueλt, we obtain the weak formulation of the characteristic equation associated with (1):

λ2 (u, χ) + k λ ( u, χ) + ( u, χ) Ω Ω ∇ ∇ Ω ∇ ∇ Ω 2 (53) + λ u, χ + kΓαλ Γu, Γχ + kΓ Γu, Γχ = 0 h iΓ1 h∇ ∇ iΓ1 h∇ ∇ iΓ1 660 NICOLAS FOURRIER AND IRENA LASIECKA

After an appropriate meshing, with N nodes, of the domain Ω into rectangles, define 1 1 Sh h as the classical family of finite-dimensional subspace of H (Ω) H (Γ1), { } Γ0 × where h denotes the maximal size of the rectangle’s sides. A typical element of Sh is a piecewise linear function which can be uniquely expressed as a linear combination N of pyramid functions Φj j=1 Sh. We refer the reader to Thomée[41], or any classical textbook describing{ } the⊂ Galerkin method for more details. After the space e discretization, with uj the value of u at the node j we set: N u(x, y) u (x, y) = ueΦ , where ue = ue, ue, ..., ue T ≈ h j j { 1 2 N } j=1 X By choosing χ = u, the discretized weak form (53) leads to N algebraic equations, which once gathered can be rewritten in the following matrix form: 2 2 e λ M + kΩλK + K + λ MΓ + kΓαλKΓ + kΓKΓ u = 0 (54) where M, K, MΓ, KΓ are the stiffness and mass matrices of size N N characterized by: ×

Kij = ΦjΦi dxdy Stiffness Matrix in Ω Ωe Mij = Φj Φi dxdy Mass Matrix in Ω RΩe M ∇Γ1 ∇Γ1 G = Φ Φ ds Stiffness Matrix in Γ1 ij RΓe i j K Γ1 Γ1 G = ΓΦ ΓΦ ds Mass Matrix in Γ1 ij RΓe ∇ i ∇ j The approximated eigenvaluesR of (1) are obtained by determining numerically the values of λ such that the determinant of (54) is null. Recall our initial model (1): u k ∆u ∆u = 0 in Ω tt − Ω t − u ∂ (u + k u ) k ∆ (αu + u) = 0 in Γ  tt ∂ν Ω t Γ Γ t 1 u =− 0 − in Γ  0  u(0, x) = u0, ut(0, x) = u1 on Ω  We will present the results for the different cases provided by this model as a function of the values taken by kΩ, kΓ and α. Assume that the damping condition holds (2), it guarantees not only the generation of the strongly continuous semigroup of contraction (Theorem 3.1), but also the spectral property σ( ) iR = (Theorem 3.5), which will be confirmed by all the pictures. A ∩ ∅ 9.1. Wave equation with strong damping in the interior and dynamic boundary condition without Laplace-Beltrami term - Figure3. Consider the following set-up: kΓ = 0, kΩ > 0, i.e. we have a wave equation with a strong damping in the interior, with no Laplace-Beltrami term on the boundary. Then, the semigroup is analytic on p for 1 p by theorem 3.4 and exponentially stable by the spectral propertyH from theorem≤ ≤ 3.5 ∞. One could also obtain the exponential stability by theorem 3.2. First of all, the following picture confirms that there is no eigenvalue on the imaginary axis. Also the spectrum is similar to the one described by Chen and Triggiani in [11], i.e., if λn n∈Z denote the eigenvalues for a strong damped wave equation with Dirichlet{ boundary} conditions, then the +,− α 2 −1 authors explicitely derived the following formula λ = µ k k µn , n n − Ω ± Ω − ∞ q µ n=1 are the eigenvalues of the laplace operator in 2D as n . This pattern {will} come again in some of the following pictures. → ∞ STRONG DAMPED WAVE EQUATION WITH DBC 661

Imaginary Axis

kΩ = 0.12

α = 0 5i

kΓ = 0

Real Axis −14 0

u k ∆u ∆u = 0 on Ω tt − Ω t − u + δ(kΩut+u) k ∆ (αu + u) = 0 on Γ  tt δη − Γ Γ t 1  u = 0 on Γ0 

Figure 3. Specturm of (1) with kΓ = 0, kΩ > 0.

9.2. Wave equation with strong damping both in the interior and on the boundary - Figure4. Consider the following set-up: kΓ, kΩ, α > 0. Then the semigroup associated with this model is analytic on by theorem 3.4 and exponentially stable by the spectral property from theoremH 3.5 associated with Pazy’s theoremA. Note that theorem 3.2 also provides the exponential stability. We obtain a similar spectrum to the previous one. On Figure4 , we present another aspect of the study, observing dynamically the impact of the coefficient on the spectrum. It shows that the structural damping (kΩ∆ut and α∆Γut) affects the radius of the circle, it is actually inversely proportional to kΩ and kΓ, i.e., as the coefficient kΩ and kΓ increases, the circle shrinks. This is the exact pattern described for a wave equation with 0-Dirichlet boundary conditions (see [11]).

Imaginary Axis

k = 0 0.24 Ω → α = 0 0.24 8.3i → kΓ = 1

Real Axis −15 0

u k ∆u ∆u = 0 on Ω tt − Ω t − u + δ(kΩut+u) k ∆ (αu + u) = 0 on Γ  tt δη − Γ Γ t 1  u = 0 on Γ0 

Figure 4. Specturm of (1) with kΓ, kΩ, α > 0 as kΩ, kΓ increases from 0 to 0.3. 662 NICOLAS FOURRIER AND IRENA LASIECKA

9.3. Wave equation with strong damping only on the boundary - Figure 5. The scenario corresponds to Theorem 3.6( kΩ = 0 and kΓ, α > 0). Since the semigroup governing this problem is a C0-semigroup of contraction on (Theorem 3.1) and the intersection between its spectrum and the imaginary axisH is empty (3.5), then, by application of Arendt-Batty theorem (TheoremB), the strong stability immediately follows. Figure5 conﬁrms this result. Indeed,we observe one new component of the spectrum spreading asymptotycally along the imaginary axis, with Re(λ) < 0, λ σ( ). We know that boundary dissipation in the absence of of inertial terms∀ on∈ the boundaryA leads to exponential stability of the semigroup. However, inertial boundary terms added to the dynamics destroy exponential stability and boundary damping alone leads to strong stability only. This fact was ﬁrst observed by Littman and Markus [26].

Imaginary Axis

kΩ = 0

α = 1 18i

kΓ = 0.0005

Real Axis −0.005 0

u k ∆u ∆u = 0 on Ω tt − Ω t − u + δ(kΩut+u) k ∆ (αu + u) = 0 on Γ  tt δη − Γ Γ t 1  u = 0 on Γ0 

Figure 5. Specturm of (1) with kΓ, α > 0 and kΩ = 0.

9.4. Wave equation with strong damping only in the interior - Figure6. Consider the following set-up: kΓ, kΩ > 0 and α = 0. Then the semigroup associated with this model is still not analytic but we recover the exponential satbility property by theorem 3.2. We observe that the spectrum in the absence of boundary damping (Figure6 ) has an additional component: a parabola. We indicated that the parabola is of the form Imλ = a b Reλ 2, again this confirms that the semigroup can not be analytic,| but| must− have× | some| smoothing property from paraboliticy. We expect the semigroup to in a class between differentiability and analyticity, and related to the theory of Gevrey Semigroups [39]. As for characterization of analyticity, Gevrey’s regularity is described in terms of the bounds on all derivatives of the semigroup: Definition 9.1 (Gevrey Semigroup). Let T (t) be a strongly continuous semigroup on a Banach space X and let δ > 1. We say that T (t) is of Gevrey class δ for t > t0 if T (t) is infinitely differentiable for t (t , ) and for every compact (t , ), ∈ 0 ∞ K ⊂ 0 ∞ and each θ > 0, there exists a constant C = Cθ,K such that

T (n)(t) Cθn(n!)δ, t and n 0, 1, 2, ... ≤ ∀ ∈ K ∈ { }

STRONG DAMPED WAVE EQUATION WITH DBC 663

Besides other characterizations, the author provided some sufficient conditions for semigroups to be of Gevrey class. Theorem E ([39]). Let T (t) be a strongly continuous semigroup satisfying T (t) Meωt. Assume one of the following holds: k k ≤ Suppose that for some γ satisfying 0 < γ 1: • ≤ lim sup β γ R(iβ; ) = C < β→∞ | | k A k ∞ Suppose that: • lim tδ T 0(t) = 0 t↓0 k k 1 Then T (t) is of Gevrey class δ for t > 0 (for every δ > γ for the first item). The resolvent characterization relies on the contour shape formed by the eigenvalues whose imaginary parts tend to infinity. More precisely, the degree of the described curve is δ. Back to our problem, we identify on Figure6 such a parabolic component which suggests that the semigroup should of Gevrey class δ = 2. This remark has not yet been proved, but the picture suggests further investigation in this direction should lead to a positive conclusion. It is also important to note from TheoremE that for γ = 1 the semigroup is analytic. This class of semigroup offers an intermediate level between analytic semigroup and differentiable semigroup. Also, while a differentiable semigroup is not stable under bounded perturbation, as it is demonstrated by Renardy in [38], if B and one of the characterizations from theoremE holds, then not only is Gevrey∈ H but also + B. This can be seen by using a similar argument as for perturbationA of analyticA semigroup by bounded operator (see [37, Theorem III.3.2.1]). One could summarize the connection between these different classes of semigroup with the following tree:

Analytic T heoremE Gevrey Differentiable → → → S. Taylor and W. Littman used Gevrey’s regularity to study smoothing properties in the context of plate and Schrodinger equations [28, 31, 40]. We also refer the reader to: [22, 23, 30, 31] for other applications of Gevrey regularity. 9.5. Wave equation with frictional damping in the interior and strong damping in the interior - Figure7. Finally, we conclude by a slight modiﬁcation of our initial model (1) by replacing the viscoelastic damping ∆ut in the interior by a frictional term ut. Thus, we now consider the model:

utt + cΩut ∆u = 0 x Ω, t > 0 u = 0 − x ∈ Γ , t > 0  ∈ 0 (55) u + ∂ u k ∆ (αu + u) = 0 x Γ , t > 0  tt ∂ν Γ Γ t 1  u(0, x) = u−, u (0, x) = u x ∈ Ω 0 t 1 ∈ The proof for theorem 3.1 would also work to show the generation of the C -  0 semigroup associated with problem (55). Also, a similar version of the proof of 2 theorem 3.2 is also working by noting that the dissipation term ut Ω in (14) is 2 |∇ | replaced by ut Ω and during the reconstruction of the energy with the multiplier u, it is not necessary| | anymore to use Poincar´e’sinequality as in (20) and (21) since we 664 NICOLAS FOURRIER AND IRENA LASIECKA

Imaginary Axis Im λ = 21.6(Re λ )2 58.3(Re λ ) + 1.14 | | | | − | | kΩ = 0.02

α = 0 20i

kΓ = 1

Real Axis −7.3 0

u k ∆u ∆u = 0 on Ω tt − Ω t − u + δ(kΩut+u) k ∆ (αu + u) = 0 on Γ  tt δη − Γ Γ t 1  u = 0 on Γ0 

Figure 6. Specturm of (1) with kΓ, kΩ > 0 and α = 0. recover immediately the desired dissipation u 2 . Therefore, this model generates | t|Ω an exponentially stable C0-semigroup of contractions

Imaginary Axis cΩ = 1

cΓ = 0

kΩ = 0 10i α = 0.1

kΓ = 1

Real Axis −7.9 0

u + c u k ∆u ∆u = 0 on Ω tt Ω t − Ω t − u + c u + δ(kΩut+u) k ∆ (αu + u) = 0 on Γ  tt Γ t δη − Γ Γ t 1  u = 0 on Γ0 

Figure 7. Specturm of (55) with cΩ, kΓ, α > 0.

9.6. Wave equation with frictional damping in the interior and high order dynamic boundary condition without Laplace-Beltrami term - Figure8. Keeping the previous model (55) in the absence of Laplace-Beltrami term on the boundary (kΓ = 0, cΩ > 0). Then, we have a C0-semigroup of contraction by a similar argument to theorem 3.1 which is only strongly stable by 3.5.

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Imaginary Axis cΩ = 0.5

cΓ = 0

kΩ = 0 14i α = 0

kΓ = 0

Real Axis −0.15 0

u + c u k ∆u ∆u = 0 on Ω tt Ω t − Ω t − u + c u + δ(kΩut+u) k ∆ (αu + u) = 0 on Γ  tt Γ t δη − Γ Γ t 1  u = 0 on Γ0 

Figure 8. Specturm of (55) with kΓ = 0, cΩ > 0.

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