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Chapitre 5 : Convergence de l’´equation desondes amorties `a l’int´erieur vers l’´equationdes ondes amorties sur le bord1 IntroductionThis article is devoted tothe comparison ofthe dynamics ofthe wave equation dampedin the interior of the domain Ω with the dynamics of the wave equation damped on theboundary of Ω, when the interior damping converges to a Dirac distribution supported bythe boundary.One of the physical motivation is the following. We consider a soundproof room, wherecarpetcoversallthewalls.Thissituationismodeledasfollows.LetΩbeasmoothboundedd ∞domain ofR (d =1,2 or 3) and letγ be a non-negative function inL (∂Ω) (the effectivedissipation of the carpet at a point of the wall). The propagation of waves in the room ismodeled by the wave equation damped in the boundaryu (x,t) = (Δ−Id)u(x,t)+f(x,u(x,t)) , (x,t)∈ Ω×R tt +∂u(x,t)+γ(x)u (x,t) = 0 , (x,t)∈∂Ω×R (1.1)t +∂ν 1 2(u,u ) =(u ,u )∈H (Ω)×L (Ω)t |t=0 0 1Notice that, in this model, the waves are not dissipated in the interior of the room butinstantaneously damped at each rebound on the walls. This corresponds to a ponctualdissipation of the form γ(x) ⊗ δ , where δ is the Dirac function supported byx∈∂Ω x∈∂Ωthe boundary. Of course, this is an approximation of the reality, as the carpet has somethickness. Thus, we can model more precisely the propagation of waves in the soundproofroom by the equationu (x,t)+γ (x)u (x,t) =(Δ−Id)u(x,t)+f(x,u(x,t)) , (x,t)∈ Ω×R tt n t +∂ u(x,t)= 0 , (x ...

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• Publié par: Phyon
• Nombre de lectures: 9
• Langue: English

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Chapitre5:Convergencedel’e´quationdes ondesamorties`al’inte´rieurversl’´equation des ondes amorties sur le bord

1 Introduction This article is devoted to the comparison of the dynamics of t he wave equation damped in the interior of the domain Ω with the dynamics of the wave eq uation damped on the boundary of Ω, when the interior damping converges to a Dirac distribution supported by the boundary. One of the physical motivation is the following. We consider a soundproof room, where carpet covers all the walls. This situation is modeled as follows. Let Ω be a smooth bounded domain ofR d (d= 1,2 or 3) and letγbe a non-negative function inL ∞ (∂Ω) (the effective dissipation of the carpet at a point of the wall). The propaga tion of waves in the room is modeled by the wave equation damped in the boundary   ( u tt ∂u ∂ν ( ( x , , x ) t t | ) )+ 0 = γ ( ( = Δ( u x) − 0 , u t I ( u d x 1 ) , ) u t) ∈ (x = , H t 1 0 ) ( , + )Ω f × (x L , 2 u Ω( (x ) , t)),( ( xt t , ,x ) ) ∈ ∈∂ Ω Ω × × R R + + (1.1)  u, u t t= Notice that, in this model, the waves are not dissipated in th e interior of the room but instantaneously damped at each rebound on the walls. This co rresponds to a ponctual dissipation of the formγ(x)⊗δ x∈∂Ω , whereδ x∈∂Ω is the Dirac function supported by the boundary. Of course, this is an approximation of the real ity, as the carpet has some thickness. Thus, we can model more precisely the propagatio n of waves in the soundproof room by the equation   u tt (x, t) +γ n (x)u t (x, t) = (Δ−I d 2 ) ( u( Ω) x, t) +f(x, u(x, t)),(x, t)∈Ω×R + (1.2) ∂ ∂ ν u(x, t) = 0,(x, t)∈∂Ω×R +  (u, u t ) |t=0 = (u 0 , u 1 )∈H 1 (Ω)×L whereγ n  oodfunction, which is positive on a small neighborhis a bounded  of∂Ω and vanishes elsewhere. The purpose of this paper is to study the relevance of the mode l equation (1.1), that is to understand in which sense the dynamics of Equation (1.2) converge to the ones of Equation

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(1.1) whenγ n converges toγ ∞ =γ(x)⊗δ x∈∂Ω in the sense of distributions. This paper is also an opportunity to present in a different way some class ical proofs on stability of gradient Morse-Smale systems. Both equations have been extensively studied, we refer for example to [8], [10], [14], [23], [40] and [45] for the wave equation with internal damping (1.2) ; and [9], [11], [29], [31], [32], [44] and [46] for the wave equation with boundary damping (1. 1). However, the conver- gence of the dynamics of Equation (1.2) to these of Equation ( 1.1) has apparently not yet been studied. The only work in this direction is the converge nce of the internal control of the wave equation towards boundary control in the one-dimensional case (see [13]). In this paper, we have chosen to focus on the convergence of the compa ct global attractor of (1.2) to the one of (1.1), when they exist, and on the comparison of t he respective dynamics on them. Indeed, the compact global attractor, which consists of all the globally bounded solutions onR, is somehow representative of the dynamics of the equation. We note that the study of convergence of attractors for other less regula r perturbations is classical ; the main tools can be found for example in [19], [3], [4] and [41]. We introduce the spacesX=H 1 (Ω)×L 2 (Ω) andX s =H 1+s (Ω)×H s (Ω) (s∈R). In the general case, we are able to prove results similar to the foll owing one. 2 Let Ω be a smooth bounded domain ofR, letγ ∞ =δ x∈∂Ω andγ n (x) =nif dist(x, ∂Ω)< 1nand 0 elsewhere. Letf∈ C 2 (Ω×R,R) be such that sup x∈Ω lim sup |u|→+∞ f(u u x) <0 and that there exist two constantsC >0 andp∈R + so that|f ′ u ′ u (x, u)|+|f x ′′ u (x, u)|< C(1 +|u| p ) for (x, u)∈Ω×R. Theorem 1.1.LetΩ,γ n ,γ ∞ andfbe as described above. Then, Equations (1.1) and (1.2) have compact global attractorsA ∞ andA n respectively. Moreover, the union of the attrac- tors(∪ n∈N∪{+∞} A n )is bounded inXand the attractors(A n )are upper-semicontinuous at A ∞ inX −s , for anys >0, that is, sup A n U ∞ i ∈ nf A kU n −U ∞ k X −s −→0 U n ∈ ∞ If all the equilibrium points of (1.1) are hyperbolic, the at tractors(A n )are lower- semi- continuous inXatA ∞  ted. Moreover, the upper and lower semicontinuity can be estima in the sense that there existsδ >0such that max  U s n u ∈ p A n U ∞ in ∈ f A ∞ kU n −U ∞ k X −s  in; sup ∈ f A n kU ∞ −U n k X  ≤ n 1 δ  U ∞ ∈A ∞ U n In general, we cannot prove upper-semicontinuity inXbecause the perturbation is too singular. LetA n andA ∞ be the linear operators associated respectively with the equations (1.2) and (1.1). The perturbation is not regular in the sense thateA n tdoes not converge to e A ∞ t inL(X). However, we can prove that, in general,A n −1 converges toA − ∞ 1 inL(X) and

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that this convergence of the inverses implies convergence of the trajectories inX −s for any initial data inX, and convergence of the trajectories inXif the initial data (u 0 , u 1 ) are bounded in a more regular spaceX s (s >0). The proof of the lower-semicontinuity inXuses as main arguments the gradient structure of (1.1) and (1.2), as well as the convergence of the local unstable manifolds of the equilibria. To prove this property, we identify the local unstable manifolds with local strongly unstable manifolds and show the continuity of these manifolds with re spect to the parametern. Although our perturbation is irregular, we can prove lower-semicontinuity inXdue to the regularity of the local unstable manifolds of the equilibria of the limit problem. The upper-semicontinuity instead cannot be shown inXin general. Indeed, we know that the union∪ n A n is bounded inXknow if it is bounded in a more regular, but we do not spaceX s . Thus, for initial data in∪ n A n , we are able to compare the trajectories only in the norm ofX−s. To prove upper-semicontinuity inX, we need to bound∪ n A n inX s for somes >0. The main way to prove this property is to show a uniform decay rate for the semigroups, that is that there exist constantsM >0 andλ >0 such that, for allU∈Xandt≥0, we have ∀n∈N,ke A n t Uk X ≤M e −λt kUk X (1.3) Such estimate is well-known for fixedn. However, the methods for proving the exponential decay for fixednoften give constantsMandλdepending onkγ n k L ∞ , or are based on a contradiction argument. Thus, they are not adaptable to the proof of a uniform estimate in the case of our irregular perturbation, wherekγ n k L ∞ goes to +∞. In dimension two and higher dimension, the uniform bound (1.3) is not known to hold, except for some very particular examples presented here. In the one-dimensiona l case, we give necessary and sufficient conditions for (1.3) to hold. The proof uses a multi plier method and is inspired by [13] and [23] (other methods are also possible, see the res ult of [2] in the appendix). Thus, in dimension one, we can show a more precise result, whi ch is typically the following. Let Ω =]0,1[,γ ∞ = 2δ x=0 andγ n (x) = 2nifx∈]0, 1 n [ and 0 elsewhere. Letf∈ f(xu) e that we do not choose C2([0,1]×R,R) be such that sup x∈Ω lim sup |u|→+∞ u <0. Notic γ ∞ =δ x=0  isfybecause, with this dissipation, Equation (1.1) does not sat the backward uniqueness property. Without backward uniqueness result, we cannot properly define the Morse-Smale property (see [11] and the remarks preceding Th eorem 2.12). Theorem 1.2.LetΩ,γ n ,γ ∞ andfbe as described above. Then, Equations (1.1) and (1.2) have compact global attractorsA ∞ andA n respectively. Moreover, the union of the attractors(∪ n∈N∪{+∞} A n )is bounded inXsfors∈]0,12[. As a consequence, the attractors A n are upper-semicontinuous atA ∞ in the space X. If all the equilibrium points of (1.1) are hyperbolic, then t he sequence of attractors(A n )is

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continuous inXin the sense that there existsδ >0such that max  U n su ∈ p A n U ∞ i ∈ nf A ∞ kU n −U ∞ k X ; sup in ∈ f A n kU ∞ −U n k X  ≤ 1 δ  U ∞ ∈A ∞ U n n In dimension one, we can even go further and compare the dynam ics on the attractors A n andA ∞ the notion of equivalence of phase-. A part of this comparison is described by diagrams. LetS(t) be a gradient dynamical system which admits a compact global attractor with only hyperbolic equilibrium points. IfEandE ′ are two equilibrium points ofS(t), we say thatE≤E′if and only if there exists a trajectoryU(t)∈ C0(R, X) such that limU(t) =E ′ and t li + m ∞ U(t) =E  t→−∞ → The phase-diagram ofS(t) is the above oriented graph on the set of equilibria. Two pha se- diagrams are equivalent if there exists an isomorphism between the set of equilibria, which preserves the oriented edges. It is proved in [19], [37] and [38] that the stability of phase -diagrams is related to the Morse-Smale property. We recall that a gradient dynamical s ystemS(t) has the Morse- Smale property if it has a finite number of equilibrium points which are all hyperbolic and if the stable and unstable manifolds of these equilibria int ersect transversally. The result of [19] says that ifS 0 (t) is a  erty,dynamical system, which satisfies the Morse-Smale prop and ifS ε (t) is a “regular” perturbation ofS 0 (t) such that the compact global attractors ofS ε (t) are upper-semicontinuous atε= 0, thenS ε (t) satisfies the Morse-Smale property forεsmall enough and its phase-diagram is equivalent to the one o fS 0 (t). Unfortunately, our perturbation is not regular enough for a direct applicat ion of [19]. However, using the smoothness of the attractors, we can adapt the proof of [19] t o show the following result. Theorem 1.3.LetΩ,γ n ,γ ∞ andfas in Theorem 1.2. If the dynamical system gene-be rated by (1.1) satisfies the Morse-Smale property, then, fornlarge enough, the dynamical system generated by (1.2) satisfies the Morse-Smale property and its phase-diagram is equi- valent to the one of (1.1). Moreover, there exists a homeomor phismhdefined fromA n into A ∞ which maps the trajectories ofS n (t) |A n onto the trajectories ofS ∞ (t) |A ∞ sererpivgn the sense of time. We notice that (1.1) satisfies the Morse-Smale property for a generic non-linearityf (see [26]). We also enhance that we give a proof of Theorem 1.3 presented in a way, which is different from [19], and, which extensively uses the gradi ent structure of (1.1) and (1.2). Of course, in this paper, we do not only consider the particul ar situations of Theo- rems 1.2 and 1.1, but more general cases. The general frame, t he main hypotheses and the main results are stated in Section 2. The abstract result of c onvergence for semigroups of contractions and the study of the convergence of the trajectories of Equation (1.1) to those

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of Equation (1.2) are given in Section 3. Continuity of the lo cal unstable manifolds and of part of the local stable manifolds as well as stability of p hase-diagrams are studied in Sections 4 and 5 respectively. In Section 6, we give concrete conditions under which the inequality (1.3) holds. In Section 7, we describe examples o f applications. Finally, in the Appendix, we state the above-mentionned result of [2] and st udy another one-dimensional case.

2 Setting of the problem and main results In this section, we first introduce the notation. We immediat ely prove a first result of convergence, without which nothing can be done. This leads t o a condition, which will be implicitely assumed in all what follows. Finally, in the las t part of this section, we put together the main hypotheses, which will be used, and state t he most important results.

2.1 The abstract frame We introduce an abstract frame for Equations (1.1) and (1.2) . This has two purposes. The first one is to give results, which concern a larger family of equations than (1.1) and (1.2) (for example, other boundary conditions can be chosen ). The second advantage of the abstract setting is to gather Equations (1.1) and (1.2) i nto a common frame, which makes the comparison easier. Let Ω be a smooth bounded domain ofR d (d= 1,2 or 3) and letω N be a non-empty smooth open subset of∂Ω. We denote byω D the largest open subset of∂Ω\ω N . Ifω D 6=∅, we setB=−Δ BC where Δ BC is the Laplacian with Neumann boundary condition onω N and Dirichlet condition onω D . Ifω N covers the whole boundary, we set B=−Δ N +I dwhere Δ N is the Laplacian with Neumann boundary condition. In all the cases,Bis a positive self-adjoint operator fromD(B) intoL2(Ω). Let (λ k , ϕ k ) be the set of eigenvalues ofBand corresponding eigenvectors normalized in L2(Ω). We denoteD(Bs2) the Hilbert space D(B s2 ) = n u= X c k ϕ k kuk 2 D(B s2 ) = X |c k | 2 λ s k <+∞ o  We notice that fors∈[0,12[,D(Bs2) =Hs(Ω) and fors∈]12,54],D(Bs2) =Hs(Ω)∩ {u∈H s (Ω)u |ω D = 0}(see Proposition 2.1). For largers, the domain ofB s2 can be less simple due to the regularity problem induced by mixed bounda ry conditions. We set 2 X=D(B 12 )×L(Ω),

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endowed with the product topology. We also setX s =D(B (1+s)2 )×D(B s2 ). Letγbe a non-negative function inL ∞ (ω N ), which is positive on an open subset ofω N . We set γ ∞ (x) =γ(x)δ x∈ω N . Let (γ n ) n∈N be a sequence of non-negative functions inL ∞ (Ω), which are positive on an open subset of Ω and which converge toγ ∞ in the sense of distributions, that is that ∀ϕ∈ C 0 ∞ (R d ), Z Ω γ n ϕ−→ Z γ ∞ ϕ= Z ω N γϕ  For eachn∈Nintroduce the linear continuous operator Γ, we n , defined fromD(B 12 ) into D(B 12 ) by Γ n =B −1 (γ n ). We also introduce the operator Γ ∞ defined fromD(B 12 ) into D(B 12 ) by  (Δ−κI d)Γ ∞ u Ω on= 0   Γ ∞ u= 0 onω D ∀u∈D(B 12 ),Γ ∞ uis the solution of ν ∂ ∂ Γ ∞ u=γ(x)uonω N ,(2.1) whereκ= 1 ifω D =∅andκ= 0 if not. We remark that ∀n∈N,∀ϕ, ψ∈D(B 12 ), <Γ n ϕ|ψ > D(B 12 ) = Z Ω γ n ϕψ , and ∀ϕ, ψ∈D(B 12 ), <Γ ∞ ϕ|ψ > D(B 12 Z ∂Ω ) =γϕψ  We set 0 =  s 0 = 14 ford= 3  (2.2) s s 0 = 12 ford= 1 ord= 2 Proposition 2.1.For allε >0,s∈[0, s 0 [andn∈N∪ {+∞}, the operatorΓ n can be extended to a continuous linear operator fromD(B ε+14 )intoD(B (1+s)2 ). In particular, Γ n is a compact non-negative selfadjoint operator fromD(B ε+14 )intoD(B 12 ). Proof :The proposition follows from the regularity properties of the operatorB. Ifω N ∩ ω D =∅, then the regularity is clear sinceD(B (1+s)2 ) ={u∈H 1+s (Ω)u |ω D = 0}ifs <12 for anyd. If we have mixed boundary conditions withω N ∩ω D 6=∅, then the regularity is more difficult to obtain. In dimensiond= 2 (resp.d= 3), we refer to [16] (resp. [12]).

For alln∈N∪ {+∞}, letA n be the unbounded operator defined onXby ∀  u v  ∈X, A n  u v  =  −B(u v + Γ n v)  , D(A n ) =   u v  ∈X  v∈D(B 12 ) andu+ Γ n v∈D(B)  

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We enhance that, ifnis finite,A n is the classical wave operator ∀n∈N, A n =  0 B− Id γ n  , D(A n ) =D(B)×D(B 12 ) − Using the Hille-Yosida theorem, one shows that the operatorA n generates a linearC0−se- migroupe A n t of contractions (see [29] forn= +∞, see also [26] for a proof in the given abstract frame). In particular,A n is dissipative since ∀U= (u, v)∈D(A n ) A, < n U|U > X =−<Γ n v|v > D(B 12 ) ≤0(2.3) ForU= (u, v), we set F(U) =  f(x 0 , u)  (2.4) We are interested in the convergence of the following family of equations, whenngoes to +∞  U U t |t= = 0 A = n U U 0 + ∈ F X (U) (2.5) We first introduce conditions so that the above equations are be well-posed. In the whole paper, we assume that the non-linearityfsatisfies the following hypothesis. (NL)f∈ C 2 (Ω×R,R) and if the dimension is d=2 there existC >0 andα≥0 such that |f u ′′ u (x, u)|+|f ′′ x (x, u)| ≤C(1 +|u| α ) u d=3 there existC >0 andα∈[0,1[ such that |f u ′′ u (x, u)| ≤C(1 +|u| α ) and|f ′ u ′ x (x, u)| ≤C(1 +|u| 3+α ) Since the regularity offis not the main purpose of this paper, we choose to state Hypo- thesis (NL) in a simple but surely too strong way. For example , the conditionf∈ C2could be relaxed to the conditionf∈ C 1 iw¨HhtdeolonrcnutisdoutavirevicenaseW.assualsome an exponential growth rate for the non-linearity ifd= 2 (see [21] or [5]). We notice that, for most of our results, weaker hypotheses onfare sufficient. For example, the critical case of a cubic non-linearity in dimensiond= 3 is studied in [27]. To obtain global existence of solutions and existence of a co mpact global attractor, we also need to assume a dissipative condition forf, for example, (Diss)sup lim sup f(x, u) <0 x∈Ω |u|→+∞ u Classical Sobolev imbeddings (see for example [1]) show tha t Hypothesis (NL) implies the following properties (see Chapters 4.7 and 4.8 of [17] for a p roof).

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Lemma 2.2.Assume  number tivethat Hypothesis (NL) holds. Then, there exists a posip such that for anyu,vinH1(Ω), we have kf(x, u)−f(x, v)k L 2 ≤C(1 +kuk p H 1 +kvk p )ku−vk H 1  H 1 Moreover, ifBis a bounded set ofH 1 (Ω), then{f(x, u)|u∈ B}and{f u ′ (x, u)v|(u, v)∈ B 2 } are bounded subsets ofH σ (Ω), whereσ∈]0,1[whend= 1ord= 2andσ∈]0, 1− 2 α [when d= 3. In addition, we have ∀u∈ B,kf(x, u)k H σ ≤C σ kuk H 1 andkf ′ u (x, u)vk H σ ≤C σ kvk H 1 , where the constantC σ depends onσ, except ifd= 1. In particular,F: (u, v)∈X7→(0, f(x, u))is of classC l 1 oc 1 (X, X)and is a compact and Lipschitz-continuous function on the bounded sets ofX. Using a classical result of local existence (see [39], Chapt er 6, Theorem 1.2), we de- duce from Hypothesis (NL) that for eachn∈N∪ {+∞}, Equation (2.5) generates a local dynamical systemS n (t) onX. Proposition 2.3.Iffsatisfies (NL), then for allM >0andK >0, there exists a time T >0such that, for alln∈N∪ {+∞}andU 0 withkU 0 k X ≤M, Equation (2.5) has a unique mild solutionU n (t) =S n (t)U 0 ∈ C 0 ([0, T], X), which satisfies ∀t∈[0, T],kU n (t)k X ≤M+K  Moreover, there exists a constantC >0such that for allU 0 andU 0 ′ withkU 0 k X ≤Mand kU ′ 0 k X ≤Mwe have ∀n∈N∪ {+∞},∀t∈[0, T],kS n (t)(U 0 −U ′ 0 )k X ≤Ce Ct kU 0 −U ′ 0 k X  The hypothesis (Diss) implies global existence of trajectories, that is thatS n (t) :X−→X are global dynamical systems. Proposition 2.4.Assume thatfsatisfies (NL) and (Diss). Then, for any bounded setB ofX, for anyn∈N∪ {+∞}and for anyU 0 ∈ B,S n (t)U 0 (t≥0) is a global mild solution of (2.5) and is uniformly bounded inXwith respect totandU 0 .

Proof :ForU= (u, v)∈X, we set Φ(U 1 ) 2 =kUk 2 X − Z Ω Z 0 u f(x, ζ)dζ 

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(2.6)

From (2.3) and the density ofD(A n ) inX, we deduce that the functional Φ is non- increasing along the trajectories of the dynamical systemsS n (t) (n∈N∪ {+∞}). Indeed, letU 0 ∈D(A n ) andU(t) = (u(t), v(t)) =S n (t)U 0 , we have 2 Φ(U(t 2 ))−Φ(U(t 1 )) = Z t < A n U(t)|U(t)> X dt=− Z t 1 t 2 <Γ n v(t)|v(t)> D(B 12 ) ≤0 t 1 (2.7) Hypothesis (Diss) implies that there exist two positive constantsCandsuch that f(x, u)u≤C−u 2 and Z 0 u f(x, ζ)dζ≤C−u 2 (2.8) So, for anyU 0 ∈ Band any positive timetsuch thatS n (t)U 0 exists, we have 2 1 kS n (t)U 0 k 2 X −C≤Φ(S n (t)U 0 )≤Φ(U 0 ) Sobolev imbeddings show that Φ(U 0 ) is bounded uniformly with respect toU 0 ∈ B. Thus, the trajectories cannot blow up and are defined and bounded fo r all times.

ForU(t)∈ C 0 ([0, T], X), we can also consider the trajectoryV n (t) =DS n (U)(t)V 0 of the linearized dynamical systemDS n (U) alongU, that is the solution of n V n (t) +F ′ (U(t))V  ∂ V t n V ( n ( )0 t=) =V 0 A ∈X n (t 2().9 ) Due to Lemma 2.2,W∈X→−7F ′ (U)Wis locally Lipschitzian and Proposition 2.3 is also valid forDS n (U)(t). Moreover the trajectoriesDS n (U)(t)V 0 exist for allt∈[0, T] since DS n (U)(t) is a linear dynamical system.

2.2 Convergence of the inverses verses If the inA n −1 do not converge toA − ∞ 1 , then one cannot hope any convergence result, since we cannot even ensure that a part of the spectrum of the o perators is continuous when ngoes to +∞ natural in. That is why, we immediatly show that this convergence holds situations. In the rest of the paper, this convergence of the inverses will be assumed. A simple calculation shows thatA n is invertible of compact inverse and thatA n −1 is given by ∀  v u  ∈X, A n −1  u v  =  −Γ n u u−B −1 v  (2.10) We present here a typical situation. ˜ Letθbe a bounded open subset ofR d−1 with a boundary of classC ∞ . We set Ω =]0,1[×θ.

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