DYNAMICS FOR THE DAMPED WAVE EQUATIONS ON TIME-DEPENDENT DOMAINS

. We consider the asymptotic dynamics of a damped wave equations on a time-dependent domains with homogeneous Dirichlet boundary condition, the nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. To this end, we establish the existence and uniqueness of strong and weak solutions satisfying energy inequality under the assumption that the spatial domains O t in R 3 are obtained from a bounded base domain O by a C 3 -diﬀeomorphism r ( · ,t ). Furthermore, we establish the pullback attractor under a slightly weaker assumption that the measure of the spatial domains are uniformly bounded above.


1.
Introduction. The investigation of differential equations on time-dependent domains (sometimes called non-cylindrical domains or time-varying domains, etc.) has a long and rich history, and the main motivation of this kind of problems arises naturally from applications, such as the Stefan problems [42], the behavior of particles in time-dependent potential well [24], the model of tumor growth [15], the American option pricing problems [16], the image processing [1], and the control problems [30,41,45], etc. In addition, the review article [34] shows us lots of interesting concrete examples about the problems on time-varying domains. The second motivation of the study of such issues is theoretical interest, such as [11,20,28].
In this work we study the damped wave equations on time-dependent domains. Such equations arise in a wide variety of applications, e.g., waves reflected by a moving boundary [19], the shallow-water waves [21,27,34], the control problems [14,23] and aerodynamics [38].
Let O ⊂ R 3 be a nonempty bounded and open set with smooth boundary ∂O, and r = r(y, t) a vector function especially satisfying is a C 3 -diffeomorphism for all t ∈ R. Define We shall assume that the functionr =r(x, t), wherer(·, t) = r −1 (·, t) denotes the inverse of r(·, t), satisfies r ∈ C 3 (Q τ,T ; R 3 ) for all τ < T. ( Consider the damped semilinear wave equations on time-dependent domains: where τ ∈ R, u 0 , u 1 : O τ → R, and f : Q τ → R are given. For the nonlinear term g(u), we assume that g ∈ C 1 (R, R), g(0) = 0 and there exists a constant C > 0 such that, We also assume that there exists γ ∈ [0, γ * ] and η > 0 such that where G(u) = u 0 g(s)ds, and the constant γ * ≥ 0 will be characterized later (see (83) for details).
It is well known that the variations of the spatial domains may have a significant effect on the evolution equations, such as equilibrium solutions, eigenvalues and eigenfunctions, especially the asymptotic behavior of the solutions, see, e.g., [3,4,5,6] for the recent significant progress on this subject. On the other hand, when the variation of the spatial domains are time-dependent, which is also called non-cylindrical problems, the situation may be rather complicated, including the definition of solutions, well-posedness, regularity and especially the asymptotic behavior of the solutions, etc.
In 1997, Bernardi et al. [8] studied the linear schrödinger-type partial differential equations on non-cylindrical domains by assuming a monotonicity condition on their section with respect to time and used the method of penalization due to J.L. Lions [36] to establish the existence of the (suitably defined) weak solutions. However, due to the existence of the penalty functions, it is hard to improve the regularity and yet the uniqueness of weak solutions is unknown. In subsequent work [9], they gave a partial answer by singling out a unique solution satisfying the energy equality among all the possible weak ones.
In 2008, Kloeden et al. [32] studied the dynamics of the semilinear heat equations on such non-cylindrical domains with the monotonicity condition relying on the scheme presented in [9]. Because of lacking of strong solutions, they approximate the penalty function by Steklov averages (see [25], Chapter I for details) which are more regular in time to obtain a compact D-pullback absorbing sets. In [22], the authors considered a stochastic partial differential equation of the reaction-diffusion type obtained by the same way as presented in [33]. Due to the variation of the domain, the existence of a family of measure preserving transformation has not been assumed, so the stochastic equations on such domains generate a new type dynamical systems, named "partial-random" dynamical systems. All this being said, we note that, reaction-diffusion equations as well as stochastic reaction-diffusion equations on non-cylindrical domains are essentially different from the ones that on cylindrical domains.
Besides all the aforementioned differences in dealing with parabolic equations on non-cylindrical domains, due to some peculiarities of wave equations, the situation is somewhat more complicated. For example, the uniqueness of weak solutions for the nonlinear wave equations on a non-cylindrical domain only under monotonicity condition, as pointed out by J.L. Lions (see [36], Chapter 3, Problems 11.8), remained open until now.
On the other hand, under some additional assumptions, the existence, uniqueness and long-time dynamics of solutions of equation (4) was studied by several authors. For example, under the assumption that there exists a one-to-one mapping φ : Q → Q * of class C 3 , with bounded derivatives and inverse ψ of class C 1 and satisfying (3.1) in [20], the existence and uniqueness of global weak solutions was proved by Cooper and Bardos [20]. Very recently, under the hypothesis that the lateral boundary is time-like and the domains are expanding, Ma et al. [37] established the pullback attractors of weakly damped wave equations by presented a useful compactness criterion.
In the present paper, we establish the existence of a pullback attractor to the problem (4) under the assumption that the time-varying domains obtained by a temporally continuous dependent diffeomorphic transformation of a bounded reference domain. Our work is in the spirit of [20,22,31,32,33,37], and many ideas of this article are taken from these works. However, we develop several generalisations, which can be regarded as the main features of this paper. First of all, with totally different from the heat equations (e.g., [33]), the problem (4) may be ill-posed for a general transformation, so we assume that the transformation is hyperbolic. We also construct a sufficient condition to ensure the transformation is hyperbolic in Appendix which may be of independent interest. Secondly, compared with [37], our domains are more complicated and general, which brought some intrinsic difficulties in our process, such as obtain the strong solutions. Hence, much of our effort is in proving the existence and uniqueness of strong and weak solutions in appropriate functions spaces as well as in establishing energy inequalities. Thirdly, the nonlinearity g of equation (4) has a critical growth rate, i.e., the mapping g from H 1 0 to L 2 is continuous, but not compact (see [43] and the references therein). To overcome the difficulties brought by the critical nonlinearity, usually, the socalled energy method developed by Ball [7], a decomposition technique (e.g., see [3,46]) and the "contractive function" method (see [18], [29] and [37] for further details and some extensions) have been successfully applied to prove the asymptotic compactness of the solutions. Due to the domain is time-dependent, we combine the two ideas presented in [31] (see Section 4.1) and [29] to prove the process is D-pullback asymptotic compact. In addition, this scheme are also helpful to obtain the pullback asymptotic compact of the process generated by the solutions of the equations on cylindrical domains.
The outline of our paper is given below. In the next section, we recall some preliminaries and known results related to pullback attractors for general nonautonomous dynamical systems. Moreover, function space setting and properties of functions are given. Strong and weak solutions are considered in Section 3 and 4, respectively, in particular their existence and uniqueness. In Section 5, based on the conclusions of the previous two sections, we show that the weak solutions generate a process, which is asymptotically compact in the light of established energy estimates in Section 3 and 4. This will lead to the proof of existence of pullback attractor in an appropriate framework in Section 6. Finally, we have included Appendix A, which is devoted to the discussion of the existence of hyperbolic.

Preliminaries.
2.1. Pullback attractors. We recall the main points about the theory of pullback attractors which will apply, see Caraballo et al. [12] and Carvalho et al. [13] for more details.
Considering the domain is time-dependent, we prefer to use the language of evolutionary processes rather than the cocycle formalism since the former seems to be more appropriate for our situation.
In addition, suppose D is a nonempty class of parameterized sets of the form Definition 2.1. The process S(·, ·) is said to be D-pullback asymptotically compact if the sequence {S(t, τ n )} is relatively compact in X t for any t ∈ R, anyD ∈ D, and any sequences {τ n } and {x n } with τ n → −∞, and x n ∈ D(τ n ).
Definition 2.2. A familyB = {B(t), t ∈ R} ∈ D is said to be D-pullback absorbing for the process S(·, ·) if for any t ∈ R and anyD ∈ D, there exists τ 0 (t,D) ≤ t such that for all τ ≤ τ 0 (t,D).
For each t ∈ R, and D 1 , D 2 nonempty subsets of X t , let us denote dist t (D 1 , D 2 ) the Hausdorff semi-distance defined as 3.Â is invariant, i.e., Theorem 2.4. Suppose that the process S(·, ·) is D-pullback asymptotic compact and thatB ∈ D is a family of D-pullback absorbing sets for S(·, ·). Then, the familŷ is a D-pullback attractor for S(·, ·), which in addition satisfies ∀t ∈ R: Furthermore,Â is minimal in the sense that ifĈ = {C(t); t ∈ R} is a family of nonempty sets such C(t) is a closed subset of X t and for any t ∈ R, then A (t) ⊂ C(t) for all t ∈ R.

2.2.
Functional spaces and preliminary results. In this subsection, we recall the functional spaces and notations that we will use throughout this paper.
For any 1 ≤ q ≤ ∞, we denote by L q (τ, T ; X t ) the vector space of all functions u ∈ L 1 loc (Q τ,T ) such that u(t) = u(·, t) ∈ X t a.e. t ∈ (τ, T ), and the function u(·) X· defined by t → u(t) Xt , belongs to L q (τ, T ).
For each u ∈ L 1 loc (Q τ,T ), we can extend u trivially to R 3 × (τ, T ) bŷ For any u ∈ L 1 loc (Q τ,T ), we will denote u = u t the derivative of u with respect to time t in the sense of distributions in Q τ,T , defined by where φ = ∂φ ∂t is the classical partial derivative. The following result was proved in [33]: ) and u ∈ L 2 (τ, T ; L 2 (O t )), and then the trivial extensionû belongs to H 1 (R 3 × (τ, T )), satisfies ∂û ∂xi = ∂u ∂xi (1 ≤ i ≤ 3), and its derivative with respect to time is given bŷ As that in Kloeden, Real and Sun [33], we say that a function u ∈ L 1 loc (Q τ,T ) belongs to C([τ, T ]; L 2 (O t )) if its trivial extensionû belongs to C([τ, T ]; L 2 (R 3 )) and we say that a sequence {u m } converges to u in C([τ, T ]; Similarly, we say that a function u ∈ L 1 loc (Q τ,T ) belongs to C([τ, T ]; H 1 0 (O t )) if its trivial extensionû belongs to C([τ, T ]; H 1 (R 3 )) and we say that a sequence {u m } converges to u in C([τ, T ]; From now on, we will use (·, ·) t , · t to denote the usual inner product and associated norm in L 2 (O t ).

We consider a finite time interval [τ, T ], and set
or, equivalently, 3. Strong solutions. For each τ < T , we consider the auxiliary problem Definition 3.1. (Strong solution). A function u = u(x, t) defined in Q τ,T is said to be a strong solution for problem (9) if , and the equations in (9) are satisfied almost everywhere in their corresponding domains.
The method that we use to prove the result of existence and uniqueness is based on the transformation of our problem into another initial boundary problem which has a more complicated structure defined over a cylindrical domain.
Using a suitable change of variables, consider the problem where Similarly, one can define the strong solution of (10). From the results in the previous sections and Section 3 in [33], by Proposition IX.18 in [10], one obtain that u = u(x, t) is a strong solution for problem (9) if and only if the function v(y, t) = u(r(y, t), t) is a strong solution of the problem (10).
In order to ensure the well-posedness for our problems, from now on, we shall suppose r is hyperbolic (see e.g. [20] for some extensions). In Appendix A, we construct a criteria for the existence of hyperbolic.
(ii) If u(t) is the strong solution of (9), then ∀t ∈ [τ, T ], (u(t), u (t)) satisfies the inequality of energy where κ τ,t = sup (iii) If u 1 and u 2 are two strong solutions of equation (9), denote w = u 1 − u 2 , we have the following estimate Proof. (i) Let us prove the existence and uniqueness of strong solution.
• Existence. The existence of strong solution for the equation (10) can be obtained by the Faedo-Galerkin method (see [26,36,40]), we sketch a proof for the reader , s convenience.
Defining the time-dependent bilinear form , and let 0 < λ 1 < λ 2 ≤ λ 3 ≤ · · · be the corresponding eigenvalues. Then, λ n → ∞ as n → ∞, and we can assume that and consider the finite dimensional approximate system and Noticing that g ∈ C 1 (R), then as a direct consequence of the existence and uniqueness result for ODEs, we have that for each integer m = 1, 2, · · · there exists a unique local solution v m of the form (15) satisfying equation (17) and solving (16) To show the existence of strong solution, we need some a prior estimates about v m , and we divide the proof into several steps.
Step 1. Multiply identity (16) by d k m (t), sum k = 1, · · · , m, and recall (15) to discover 1 2 Employing integration by parts formula, we get Performing simple calculations, using the condition on r, (6) and Young inequality, we deduce where C 3 only depend on r, η and γ. Applying Gronwall inequality, which along with (5) Denote where C 0 (independent of domain O, see [26] Chapter 5 for more details) satisfying By (6) and Combining (19) and (22), we can easily deduce that for some positive constants C 4 and C 5 , which are independent of m.
Step 2. Differentiate the identity (16) with respect to t, we obtain where Multiply (24) by d k m (t), sum k = 1, · · · , m, we discover 1 2 . (26) We now estimate every term on the right-hand side of (26). Firstly, by the assumption on r, we have and Integrating by parts, we find that Similarly, the seventh term on the right-hand side of (26) are bounded by Furthermore, we also have Recalling (5), (23) and applying Hölder inequality, we deduce By (27)- (35), recalling r is hyperbolic, we conclude for some positive constant C 6 independent of m.
Step 3. Recall we are taking {e k } ∞ k=1 to be the complete collection of eigenfunctions for −∆ on H 1 0 (O). Multiplying (16) by λ k d k m (t) and summing k = 1, · · · , m, we deduce Now, we hand each term as following. and On the other hand, employing integration by parts formula, Using Lemma 3.3 and Young inequality, we arrive at where C 1 and C 2 are positive constants. By Young inequality, we get and Combining the above estimates, i.e., (37)-(43), we can obtain Using this estimate in (36), we have the following by Gronwall inequality, which along with (23), we discover where C7, C8 and C9 are positive constants (independent of m).
Step 4. Under the above estimates (45), we conclude that It is now a standard matter to prove that a subsequence of {vm} ∞ m=1 , still denote {vm} ∞ m=1 , satisfying vm v weakly star in L ∞ (τ, T ; H 2 (O)), Obviously, v is the strong solution of (10).
Multiplying the above equation (46) by w , to find Similar to the proof of Step 1, we have the following estimates As usual, we deal with the nonlinear term. To do so, using Höder inequality, recalling v1, v2 ∈ L ∞ (τ, T ; H 1 0 (O)), we find that Taking into account of (47)-(51), we conclude that which implies the uniqueness on account of Gronwall inequality immediately.
(iii) Let (ui, u i ), (i = 1, 2) be the corresponding strong solutions to (u i 0 , u i 1 ), now write w = u1 − u2 and w = w + ϑw, we discover that Multiplying the above equation (70) by w, and repeating the argument used in the proof of (12) we can establish that Applying (68) and (69), we deduce (13). The proof of Theorem 3.4 is now complete.

Remark 2.
With the help of inequality (58), we can obtain the D-pullback absorbing sets (see Section 5), avoid discussing eigenvalues (see [33] and [44], also [37] for more details), and only assuming the measure of every spatial domains are uniformly bounded above. Furthermore, this method can improve the restriction on varying domains in the aforementioned works on the topic, e.g., [33], Section 7.
Remark 3. In order to ensure the existence of strong solutions, we assume the transformation r is C 3 -diffeomorphism. As a matter of fact, this condition is slightly strong, however, this matter will be pursued elsewhere.
4. Weak solutions. In this section, combining the ideas of [33], we will give a proper definition for weak solution about the wave equations on time-dependent domains. Let us denote and −∞ < τ ≤ T < ∞ be given. We say that a function u is a weak solution of (9) if and where u m is the unique strong solution of (9) corresponding to (u 0m , u 1m , f m ); Remark 4. The strong solution u is obviously a weak solution.
Then for each (u 0m , u 1m , f m ), m = 1, 2, · · · , there exists a unique strong solution u m for the following problem: Moreover, from (12), it follows Therefore, taking into account Lemma 3.5 and 3.6 in [33], we can extract a subsequence (denoted also by {u m }) such that At the same time, we have Multiplying (78) by (u m −u n ) , repeating the argument used in the proof of (35), we discover and therefore {u m } is a Cauchy sequence in C([τ, T ]; L 2 (O t )).
So by the uniqueness of the limit, (75) and (76), we find that Therefore, we can assume that g(u m ) → g(u), a.e. in Q τ,T , and then from (77), we can find χ = g(u). Finally, we will show that u is the unique weak solution of the equation (9). For any test function φ ∈ Φ τ,T , we know that u m satisfies (71). Then, using (75)-(77), and (80)-(81), by passing to the limit, we obtain that u also satisfies (71). So u is a weak solution of (9) with initial data u 0 , u 1 .
Proof. Let us fix τ < t. By definition we know that there are two sequences where u i m is the unique strong solution corresponding to the regular data Then, similarly to (35), we have Therefore, we get (84) immediately from (85), (86) and (87).
6. D λ -pullback attractor. Throughout this section, we assume that 6.1. D λ -pullback absorbing set. Let (u 0 , u 1 ) ∈ X τ , and u(t) = S(t, τ )(u 0 , u 1 ). We denote For sufficiently small ϑ > 0, from (65) and (88), we observe that Combining with (72), we discover where λ * 0 is a positive constant satisfying For each t ∈ R, we set R(t) the positive number given by Lemma 6.1. The family of setŝ is a pullback D λ -absorbing family for the solution process S(t, τ ). Moreover,B 0 belongs to D λ .
6.3. D λ -pullback attractor. From Lemma 6.2, and the fact that the sets in B 0 are closed, and the family D λ is inclusion closed, we obtain that S(t, τ ) has a D λpullback attractor, and more exactly: is the unique D λ -pullback attractor for the process S(t, τ ) belonging to D λ . In addition,Â satisfies Furthermore,Â is minimal in the sense that ifĈ = {C(t) : t ∈ R} is a family of nonempty sets such that C(t) is a closed subset of X t and lim τ →−∞ dist t (S(t, τ )D 0 (τ ), C(t)) = 0 for all t ∈ R, then A (t) ⊂ C(t) for any t ∈ R.
Remark 5. The above process and corresponding results can be applied to a variety of other dissipative equations defined on non-cylindrical domains, such as complex Ginzburg-Landau equation and Schrödinger equation, which are with a widely background in Physics (see [34] for further details, and additional references). Furthermore, the stochastic wave equations with multiplicative white noise (or additive white noise) on non-cylindrical domains, could also be treated by the presented scheme. All these problems will be reported in our subsequent papers.
Remark 6. As mentioned in our Introduction, the nonlinear wave equations on non-cylindrical domains with expanding spatial domains are difficult to deal with. Here, our results give some useful information about this issue, e.g., if the domain satisfying some regularity condition, the existence of the strong solution is objective. However, to obtain the strong solution or uniqueness of weak solutions, a new method should be developed, and a possible approach may be the one taken by Cannarsa et al. in [11]. If not, we could consider the asymptotic behavior of the solutions under the other frameworks, such as generalized semiflows [7] or evolution systems [17].

Remark 7.
With the discussion in the present paper, combing the situation and problems in [4] (see also [2,5,6]), we can assume that the perturbation of the domain is time-dependent. Hence, a natural question is: is it possible to achieve the similar results, such as spectral convergence and continuity of attractors, which are constructed in [2,4,5,6]? Recently, Pereira & Silva [39] considered the reactiondiffusion equations in a time-dependent thin domain, which is a good try for further research in this field. However, when referred to wave equations, to answer these questions, new methods and theories may be needed.
Appendix A. Appendix.
A.1. A sufficient condition. Here we construct a sufficient condition to insure that r is hyperbolic. ( ∂ri ∂t (y, t)) 2 .
Remark 8. We can easily obtain that the result is also reasonable if the space dimension n > 3.
A.2. An application. To better understand the larger scope of our results, we illustrate their application to the following case.

Remark 9.
It is easy to see that Theorem A.1 can also be applied to the case that r(y, t) = h 1 (t)y 1 , h 2 (t)y 2 , h 3 (t)y 3 , t ∈ R. E.g., see Section 2 in [37] for more details.