Time-dependent attractor of wave equations with nonlinear damping and linear memory

Abstract In this article, we consider the long-time behavior of solutions for the wave equation with nonlinear damping and linear memory. Within the theory of process on time-dependent spaces, we verify the process is asymptotically compact by using the contractive functions method, and then obtain the existence of the time-dependent attractor in H01(Ω)×L2(Ω)×Lμ2(R+;H01(Ω)) $\begin{array}{} H^{1}_0({\it\Omega})\times L^{2}({\it\Omega})\times L^{2}_{\mu}(\mathbb{R}^{+};H^{1}_0({\it\Omega})) \end{array}$.


Introduction
Let The function a(x) satis es: where α is a constant.
( . ) The nonlinear term f ∈ C (R), f ( ) = , and for some C ≥ satis es along with the dissipation condition where λ is the rst eigenvalue of the strictly positive operator A = −∆.
With respect to the memory component, as in [ − ], we assume that µ (s) ≤ −ρµ(s) ≤ , ∀s ≥ , where ρ is a positive constant. The problem (1.1) can be viewed as a description of viscoelastic solids with fading memory and dissipation due to the viscous resistance of the surrounding medium, as well as of composite materials, phase-elds, and wave phenomena [7][8][9].
When µ is a Dirac measure at some xed time instant or when it vanishes, the equation (1.1) reduces to the nonlinear damped wave equation, which has been investigated extensively by many authors. For instance, in the case that ε is a positive constant independent of time, the long-time behavior of the solution can be well characterized by using the concept of global attractors in the framework of semigroup. The existence and regular properties of the global attractor have been studied in [2,[10][11][12]. When ε is a positive constant independent of time and the forcing term h depends on time, the system is a non-autonomous wave equation, the long-time behavior of the solution can be understood in the framework of process. We refer the reader to [11,[13][14][15][16] for some speci c results involving the uniform attractor (or pullback attractors) about non-autonomous case.
When ε is still a positive constant and the nonlinear damping a(x)g(t) is either linear damping αu t or strong damping ∆u t , Conti and Pata ([17]), Borni and Pata ([18]), Pata and Zucchi ([19]) investigated the existence of global attractors for (1.1). Sun, Cao and Duan ( [20]) obtained the existence and asymptotic regularity of the uniform attractor about the non-autonomous system with strong damping, while the robust exponential attractors was scrutinized by Kloden, Real and Sun ([21]) However, provided that ε depends explicitly on time in (1.1), such as a positive decreasing function of time ε(t) vanishing at in nity, leading to time-dependent terms at functional level, these problems become more complex and interesting, because the corresponding dynamical system is still understood within the non-autonomous framework even the forcing term is independent of time, and the classical theory generally fails to capture the dissipation mechanism of the system, as mentioned in [22,23].
To circumvent these issue, in [22], Conti, Pata and Temam presented a notion of time-dependent attractor by exploiting the minimality with respect to the pullback attraction property, and constructed a su cient condition proving the existence of time-dependent attractor based on the theory established by Plinio, Duane and Temam( [23]). Meanwhile, within the new framework, the authors studied the following weak damped wave equations with time-dependent speed of propagation Besides, they proved that the time-dependent global attractor of (1.11) converged in a suitable sense to the attractor of the parabolic equation αu t − ∆u + f (u) = g(x) when ε(t) → as t → +∞ ( [24]). Successively, in [25], they continued to show the existence of an invariant time-dependent global attractor to the following speci c one-dimensional wave equation ε(t)u tt − uxx + [ + εf (u)]u t + f (u) = h, which converges in suitable sense to the classical Fourier equation.
Recently, Meng et al. investigated the long-time behavior of the solution for the wave equation with nonlinear damping g(u t ) on the time-dependent space, in which they found a new technical method verifying compactness of the process via de ning the contractive functions, see [1]. In [26], Meng and Liu also showed the necessary and su cient conditions of the existence of time-dependent global attractor borrowed from the ideas in [10]. Liu and Ma ([ ]) studied the existence of the pullback attractors for the plate equation with time-dependent forcing term on the strong time-dependent Hilbert space. Successively, exploiting the methods and framework of [22,24], Liu and Ma obtained the existence and regularity of the time-dependent attractor for the plate equation with critical growth nonlinearity, as well as the asymptotic structure in [28].
As we know, in the study of the long-time behavior, especially for attractors, obtaining certain asymptotic compactness of the solution operator is a key step. However, if the equation contains the history memory, for instance, just for our problem (1.1), it makes impossible to utilize (I − Pm)u as the test function to capture the asymptotic compactness of the solution process, that is to say, the methods introduced in [10,26] is out of action to our problem. On the other hand, because of the critical nonlinear damping, the technique of operator decomposition in [22] is not suitable to deal with (1.1) anymore. Thus, for our problem, we need to make a priori estimates to solution on a new triple solution space, and then verify the compactness of the solution process by exploiting the method of contractive function.
It is worth mentioning that we use the more weaker dissipative condition (1.8) than [1,22]; indeed, for simplicity, in which the authors made use of the dissipative condition like lim inf For convenience, hereafter, C (or c) denotes an arbitrary positive constant which may be di erent from line to line even in the same line.
The rest of this article consists of two Sections. In the next Section, we de ne some functions sets and iterate some useful lemmas. In Section 3, the existence of the time-dependent global attractor is obtained.

Preliminaries
As in Borini, Pata [18], Pata, Zucchi [19] and Dafermos [4], we introduce the past history of u, i.e. η = η t (x, s), as a new variable of the system, which will be ruled by a supplementary equation: denoting then we can rewrite (1.1) as with initial boundary conditions where Without loss of generality, set H = L (Ω) with the inner product ·, · and norm · , respectively. For s ∈ R + , we de ne the hierarchy of (compactly) nested Hilbert spaces especially, we have the embeddings H s+ → H s . For s ∈ R + , let L µ (R + ; H s ) be the family of Hilbert spaces of functions φ : R + → H s , equipped with the inner product and norm, respectively, Now, for t ∈ R and s ∈ R + , we have the following time-dependent spaces Here For every t ∈ R, let X t be a family of normed spaces, we introduce the R−ball of X t We denote the Hausdor semi-distance of two (nonempty) sets B, C ⊂ X t by: For any given ϵ > , the ϵ−neighbourhood of a set B ⊂ X t is de ned as Finally, given any set B ⊂ X t , the symbolB stands for the closure of B in X t . Now we iterate some basic notations and abstract results, which are necessary for getting our main results.

De nition 2.3. [ ] 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 > there exists a t such that  τ) is a T−closed process for some T > , which possesses a time-dependent global attractor A = {A t } t∈R , then A is invariant. Lemma 2.8. [ , ] Let g(·) satisfy condition (1.5). Then for any δ > , there exists a positive constant C δ , such

Existence of the time-dependent global attractor . Well-posedness and time-dependent absorbing set
Now we state the results about the well-posedness of system (2.2)-(2.3) which can be found in [19,20]. In fact, the existence of solution (u(t), u t (t), η t (s)) to (2.2)-(2.3) is obtained by using the standard Galerkin approximation method, which is based on Lemma 3.2 below.

Thus, the system (2.2)-(2.3) generates a strongly continuous process U(t, τ), where
with the initial data zτ = z(τ) = {u , u , η } ∈ Hτ . To prove Lemma 3.1, we rst need the following estimate: Under the assumptions (1.2)-(1.10), for any initial data z(τ) ∈ Bτ(R) ⊂ Hτ , there exists R > , such that Proof. Denote Multiplying ( . ) with u t in L and exploiting ( . ) , we achieve ( . ) By Hölder, Young inequalities, and combining with (1.10) we obtain Using that g is strictly increasing, ε(t) is decreasing, and (3.3), we have From (2.6) and Sobolev's embeddings we deduce Consequently, there exist some proper positive constant C , C and C , such that And from (3.4),(3.5) yields On the other hand, (1.6),(2.5) along with the Hölder and Young inequality imply where η > is a small enough constant, which will be determined later. Multiplying ( . ) by u t + δu and integrating over Ω, we get where therefore, we get Together with Hölder, Young, Poincaré inequalities, it follows that here we use ν − δ L λ > ν for δ < ν small enough. Thanks to (1.5)-(1.6), for any δ > , there exist C δ > such that ( . ) Collecting all the above estimates and due to (2.7), Hölder, Young inequalities, it leads to where for δ < min{ν, ρ} small enough. Then, from (3.9)-(3.12), (3.6) yields then B t is a bounded absorbing set for process U(t, τ). In addition, from the above discussion, there exists a positive constant R such that The proof for Lemma . . Let z (τ), z (τ) ∈ Hτ such that z i (τ) Hτ ≤ R, i = , , and denote by C a generic positive constant depending on R but independent of z i . We rst observe that the energy estimates in Lemma 3.2 above ensure: . Then the di erence between the two solutions with initial dataz(τ) = z (τ) − z (τ) ful lls Multiplying the above equation with ū t and integrating over Ω, we obtain From (1.4) and the strict increase of g, we have Ω a(x)(g(u t ) − g(u t ))ū t dx ≥ .