ATTRACTORS FOR WAVE EQUATIONS WITH NONLINEAR DAMPING ON TIME-DEPENDENT SPACE

. In this paper, we consider the long time behavior of the solution for the following nonlinear damped wave equation with Dirichlet boundary condition, in which, the coeﬃcient ε depends explic- itly on time, the damping g is nonlinear and the nonlinearity ϕ has a critical growth. Spirited by this concrete problem, we establish a suﬃcient and neces- sary condition for the existence of attractors on time-dependent spaces, which is equivalent to that provided by M. Conti et al.[10]. Furthermore, we give a technical method for verifying compactness of the process via contractive functions. Finally, by the new framework, we obtain the existence of the time- dependent attractors for the wave equations with nonlinear damping.

1. Introduction. In this paper, we consider the asymptotic properties of the dynamical system generated by the following wave equation with nonlinear damping on a bounded domain Ω ⊂ R 3 with smooth boundary ∂Ω, (1.1) when g(u t ) = αu t (weak damping wave equation) and obtained the existence of the time-dependent global attractor, which converges in a suitable sense to the attractor of the parabolic equation αu t − ∆u + f (u) = 0 (see [11]). Moreover, in [12], Conti, Pata considered a specific one-dimensional wave equation ε(t)u tt −u xx + [1 + εf (u)]u t + f (u) = h, they proved the existence of an invariant time-dependent attractor, which converges in a suitable sense to the classical Fourier equation.
In this paper, we will consider the system (1.1), where ε(t) depends on time explicitly, the damping g is nonlinear, and the nonlinearity ϕ has a critical exponent.
Our basic assumptions about the coefficient ε(t), the nonlinear damping g and the nonlinearity ϕ are as follows.
Whether the system is autonomous or non-autonomous, to establish the compactness of the system in some sense is the key point to prove the existence of attractors. In the case where ε(t) is a positive constant function, the difficult for obtaining compactness mainly comes from nonlinear damping and the nonlinear terms. In 2006, Sun etc. [33] firstly overcome this difficulty when system is autonomous( f did not depend on t), without assuming a large value for the damping parameter, and then they proved the existence of the global attractor for system (1.1) under the conditions that the growth order p of nonlinear damping term must be less than 5 and the growth order of nonlinear term can equal to 3, in which the exponent about ϕ is called critical exponent since the nonlinearity ϕ is not compact directly using Sobolev embedding. Later on, in [19], the author improved the result in [33] and obtained the existence of the global attractor in the case of p = 5, by estimating the boundedness of be open as we know, the main reason is that we can only get When ε(t) is not a constant function, but rather a positive decreasing function of time ε(t) vanishing at infinity, even f does not depend on t, this system is still non-autonomous, and then much more complex. In [10], the author proved the existence of the time-dependent global attractor for (1.1) where g(u t ) = αu t , the strategy of verifying asymptotic compactness for the corresponding process consists in finding a suitable decomposition of the process in the sum of a decaying part and of a compact one. However, for the nonlinear damping, it appears difficult to apply the method of [10] to deal with the compactness of the system.
Motivated by the problem and inspired by [1,24,32,34,35], we first introduce a concept about compactness for the process on time-dependent space, which we term as pullback asymptotically compact. Then we give a criterion ensuring the existence of a time-dependent global attractor under the assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets, which seems to be more suitable to deal with the model (1.1). Moreover, we propose a technique method via contractive functions on time-dependent space to verify pullback asymptotic compactness for the process. The technique to prove some compactness of systems (in autonomous case) was initiated by I. Chueshov and I. Lasiecka [7,8] in the context of wave equation with nonlinear damping, and subsequently, by A. Kh. Khanmamedov [20] in the context of Von Karman equations. Later on, Sun et.al [32,34] extended this technique to non-autonomous systems. Our aim is to extend this technique to non-autonomous systems on time-dependent space. Finally, we show the existence of time-dependent global attractor for the system (1.1) by using the new framework.
The paper is organized as follows. In Section 2, we make some preparations for our consideration; in Section 3, we introduce the concept of pullback asymptotic compactness, and develop a criteria for the existence of time-dependent global attractor; in Section 4, a theoretical technique for verifying asymptotic compactness for the process is proposed; finally, in Section 5, the existence of time-dependent global attractor for the system (1.1) is proved by using the new framework.
Throughout the paper, C denotes any positive constant which may be different from line to line even in the same line (sometimes for special differentiation, we also denote the different positive constants by C 1 , C 2 ,· · · ).
2. Preliminaries. In the following subsection 2.1 and 2.2, we review briefly the notations, some basic definitions and abstract results about processes on timedependent spaces, which will be used in considering our problems, see [10,28] for more details.
2.1. Notations. Let {X t } t∈R be a family of normed spaces, we introduce the R−ball of X t B t (R) = {z ∈ X t : z Xt ≤ R}. For any given ε > 0, the ε−neighborhood of a set B ⊂ X t is defined as We denote the Hausdorff semidistance of two (nonempty) sets B, C ⊂ X t by Given any set B ⊂ X t , the symbol B stands for the closure of B in X t .
The process U (t, τ ) is called dissipative whenever it admits a pullback absorbing family.
Definition 2.4. A time-dependent absorbing set for the process U (t, τ ) is a uniformly bounded family B = {B t } t∈R with the following property: for every R ≥ 0 there exists a t 0 = t 0 (R) ≥ 0 such that Remark 1. It is obvious that the existence of a time-dependent absorbing set implies the dissipative of the corresponding process.
(ii) A is pullback attracting, i.e. it is uniformly bounded and the limit holds for every uniformly bounded family C = {C t } t∈R and every fixed t ∈ R.
Remark 2. The attracting property can be equivalently stated in terms of pullback absorbing: a (uniformly bounded) family K = {K t } t∈R is called pullback attracting if for all ε > 0 the family {O ε t (K t )} t∈R is pullback absorbing. Theorem 2.6. The time-dependent global attractor A exists and it is unique if and only if the process is asymptotically compact, namely, the set of a finite number of points such that the family {O ε t (F t )} t∈R is pullback absorbing. The process is called totally dissipative whenever it is εdissipative for every ε > 0. Remark 3. It is apparent that an asymptotically compact process is totally dissipative, the corresponding relationship about semigroup is discussed in [4].
Remark 4. If the time-dependent global attractor A exists and the process U (t, τ ) is a strongly continuous process, then A is invariant, see more details in [10].
2.3. Some properties for nonlinear function g and ϕ. In the following, we will state some properties of the nonlinear damping function g and the nonlinearity ϕ (see [10,17,20,32,34] etc. for more details), which will be used in Section 5.
Note that condition (1.6) implies that Moreover, we review the following result.
3. Criterion for the existence of time-dependent global attractor. In this section, we firstly give the definition of pullback asymptotically compact in the timedependent space case, and then using this definition, we provide a sufficient and necessary condition about the existence of attractor on time-dependent space.
Definition 3.1. We say that a process U (·, ·) in a family of normed spaces {X t } t∈R is pullback asymptotically compact if and only if for any fixed t ∈ R, bounded sequence {x n } ∞ n=1 ⊂ X τn and any Remark 5. By Theorem 2.6 which is from [10], we know that the asymptotic compactness of the process will ensures the existence of time-dependent global attractor. However, for some concrete problem, the asymptotic compactness of the process is hard to be verified, especially, when it is hard to get higher regularity of the solutions. In [10], for the wave equation, the authors gave a very nice method to verify the necessary asymptotic compactness, they found a suitable decomposition of the process in the sum of a decaying part and a compact one to overcome the lack of regularity of the solutions, and then verified the asymptotic compactness of the process successfully. However, for our problem, due to the nonlinear damping, it seems to be difficult to apply the method of [10]. Hence, inspired by the concrete problem and the classical definition of asymptotic compactness (e.g. for the semigroup [24,35], the semi-process [26] and the process [1,34] etc.), we introduce a concept of the compactness for the process on time-dependent space, which we term as pullback asymptotically compact. Note that, pullback asymptotically compact is weaker than asymptotically compact.
is asymptotically compact, then it is pullback asymptotically compact.
Proof. If U (·, ·) is asymptotically compact, i.e., there exists a pullback attracting k . Now, by the compactness of K t , there exists a subsequence (which we relabel) y k such that lim k→∞ y k = y 0 ∈ K t . Furthermore, due to n=1 has a convergent subsequence, hence U (·, ·) is pullback asymptotically compact. Now, we provide the sufficient and necessary condition for existence of timedependent global attractor. Let U (·, ·) be a process in a family of Banach spaces {X t } t∈R . Then U (·, ·) has a time-dependent global attractor A * = {A * t } t∈R satisfying if and only if (i) U (·, ·) has a pullback absorbing family B = {B t } t∈R ; (ii) U (·, ·) is pullback asymptotically compact.
immediately by Lemma 3.2.
(Sufficient.) We will verify A * = {A * t } t∈R where for any pullback absorbing set B = {B t } t∈R is the time-dependent global attractor for the process U (·, ·).
We will accomplish the proof by three steps.
Step 1. For any fixed t ∈ R, A * t = s≤t τ ≤s U (t, τ )B τ is nonempty and compact in For any fixed t ∈ R, then for any τ n ∈ R −t and x n ∈ B τn , by the definition of the pullback asymptotic compactness of the process we know that {U (t, τ n )x n } ∞ n=1 has a convergent subsequence, without loss of generality, we assume that Then by the construction of A * t we know that y ∈ A * t , which implies that A * t is nonempty.
For any y m ∈ A * t , m = 1, 2, · · · , we will show that {y m } has a convergent sub- where ρ(·, ·) is the metric on X t . Therefore, by the assumption of the pullback asymptotic compactness of the process U (t, τ ), there exists a convergent subsequence of {U (t, τ m )x m } ∞ m=1 in X t , without loss of generality, we assume that {U (t, τ m )x m } ∞ m=1 is a Cauchy sequence in X t . Then ρ(y n , y m ) we know that {y m } ∞ m=1 is also a Cauchy sequence in X t . Moreover, from the construction we have that Step 2. The family A * = {A * t } t∈R is pullback attracting, i.e. for every uniformly bounded family C = {C t } t∈R and every t ∈ R, the limit We argue by contradiction. Assume there exists a family of uniformly bounded sets {C * t } t∈R and some t 0 ∈ R such that C τ ∈ X τ , δ t0 (U (t 0 , τ )C t0 , A * t0 ) does not tend to 0 as τ → −∞. Thus there exists ε 0 > 0 and a sequence τ n → −∞ such that For each fixed n, there exists c n ∈ C τn satisfying Since B is pullback absorbing, U (t, τ n )C τn ⊂ B t , and hence U (t, τ n )c n ∈ B t , for n sufficiently large ( such that t 1 small enough and τ n ≤ t 1 ≤ t). The sequence {U (t, τ n )c n } is relatively compact, and hence {U (t, τ n )c n } has a convergent subsequence {U (t, τ ni )c ni } such that Since U (t 1 , τ ni )c ni ∈ B, β ∈ A * t0 and this contradicts (3.2).
Step 3. Minimality, i.e. if K t is compact and pullback attracts C t for every uniformly bounded family C = {C t } t∈R and every t ∈ R, then A * t ⊂ K t . Now we will show it. ∀y ∈ A * t , then there are x n ∈ B τn and τ n ∈ R −t with τ n → −∞ such that U (t, τ n )x n → y. From the assumption K t pullback attracts B t , obviously, we have δ t (U (t, τ n )x n , K t ) → 0 as n → ∞.
At the same time, the compactness of K t implies y ∈ K t . Hence, A * t ⊂ K t .
Remark 6. Note that, from Lemma 3.2, we know that pullback asymptotically compact is weaker than asymptotically compact, hence in order to provide sufficient and necessary conditions for the existence of time-dependent global attractor, we need some dissipativity of the system. In some concrete problems, some dissipativity is easier to be obtained than some compactness in certain sense. However, it is easily to conclude that, through comparing Theorem 2.6 in [10] with Theorem 3.3 in this paper, "the process U is asymptotically compact" in [10] is equivalent to "the process U is pullback asymptotically compact (see Definition 3.1) plus U has a pullback absorbing family".

4.
A technique method for verifying asymptotically compactness. In this section, we present a technical method via contractive functions to verify the pullback asymptotic compactness on time-depend spaces. The technique to prove some compactness of autonomous system was initiated by I. Chueshov and I. Lasiecka [7,8], and then was extended to non-autonomous systems by Sun et.al [32,34]. Now, we extend this method to time-depend spaces case.
We start with a preliminary definition.
Definition 4.1. Let {X t } t∈R be a family of Banach spaces and C = {C t } t∈R be a family of uniformly bounded subsets of {X t } t∈R . We call a function φ t τ (·, ·), defined on X t ×X t , a contractive function on C τ ×C τ if for any fixed t ∈ R and any sequence where τ ≤ t. We denote the set of all contractive functions on C τ × C τ by E(C τ ).
Theorem 4.2. Let U (·, ·) be a process on {X t } t∈R and has a pullback absorbing family B = {B t } t∈R . Moreover, assume that for any ε > 0 there exist , y), ∀x, y ∈ B T , for any fixed t ∈ R. Then U (·, ·) is pullback asymptotically compact.
Proof. B = {B t } t∈R is pullback absorbing, we only need to show that U (t, τ n )x n is precompact in X t for any {x n } ∞ n=1 ⊂ B τn and {τ n } with τ n → −∞ as n → ∞. In the following, we will prove that {U (t, τ n )x n } ∞ n=1 has a convergent subsequence via diagonal methods.
Taking ε m > 0 with ε m → 0 as m → ∞. At first, for ε 1 , by the assumptions, there exist T 1 = T 1 (ε 1 ) ≤ t and φ t T1 (·, ·) ∈ E(B T1 ) such that for any fixed t ∈ R, where φ t T1 depends on T 1 . Since τ n → −∞, for such fixed T 1 , without loss of generality, we assume that τ n ≤ T 1 , and for each n ∈ N such that U (T 1 , τ n )x n ∈ B T1 . Set y n = U (T 1 , τ n )x n . Then from (4.1) we have By the definition of E(B T1 ) and φ t T1 ∈ E(B T1 ), we know that {y n } ∞ n=1 has a subsequence {y Similar to [20], we have which, combining with (4.2) and (4.3), implies that Therefore, there exists a K 1 such that By induction, we obtain that, for each m ≥ 1, there is a subsequence n k } ∞ k=1 is a Cauchy sequence in X t . This shows that {U (t, τ n )x n } ∞ n=1 is precompact in X t . Thus we complete the proof.

Time-dependent global attractor for non-autonomous wave equation.
In this section, we prove the existence of time-dependent global attractor for the wave equation with nonlinear damping by applying Theorem 3.3. First, the wellposedness of the problem (1.1) and then the corresponding process is established in 5.1; the dissipativity of the process is obtained by appropriate energy estimates in 5.2; then some a priori estimates are established which will be used to obtain the asymptotic compactness of the process in 5.3; in 5.4, the compactness of the process is verified by using the technique method presented in Section 4; the main result on the existence of the time-dependent global attractor is stated at the end of this section.
Denote the time-dependent space Note that the spaces H t are all the same as linear spaces and the norms · 2 Ht and · 2 Hτ are equivalent for any fixed t, τ ∈ R.
Moreover, we state the continuous dependence estimate for U (t, τ ) on H τ , which can be to verify the uniqueness of the solution.
Proof. Given two different initial datum z 1 (τ ), z 2 (τ ) ∈ H τ such that z i (τ ) Hτ ≤ R, i = 1, 2. By Theorem 5.3 below we know that Multiplying the above equality by 2ū t and integrating over Ω, we have Combining with (1.7) and (5.2), we have In addition, the following inequality is hold: Then we can obtain the differential inequality Applying the Gronwall Lemma on [τ, t], Ht ≤ e C(L+1) t τ 1 ε(s) ds z(τ ) 2 Hτ , the proof is completed.
This concludes the proof of the existence of the time-dependent absorbing set.