Upper semicontinuity of pullback attractors for a nonautonomous damped wave equation

In this paper, we study the local uniformly upper semicontinuity of pullback attractors for a strongly damped wave equation. In particular, under some proper assumptions, we prove that the pullback attractor {Aε(t)}t∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{A_{\varepsilon }(t)\}_{t\in \mathbb{R}}$\end{document} of Eq. (1.1) with ε∈[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon \in [0,1]$\end{document} satisfies limε→ε0supt∈[a,b]distH01×L2(Aε(t),Aε0(t))=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lim_{\varepsilon \to \varepsilon _{0}}\sup_{t\in [a,b]} \operatorname{dist}_{H_{0}^{1}\times L^{2}}(A_{\varepsilon }(t),A_{ \varepsilon _{0}}(t))=0$\end{document} for any [a,b]⊂R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[a,b]\subset \mathbb{R}$\end{document} and ε0∈[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon _{0}\in [0,1]$\end{document}.


Introduction
The theory of pullback (or random) attractors is a useful tool to study the long-time behavior of nonautonomous (or random) dynamical systems (see [1,3,7] and references therein), in which the trajectory can be unbounded as "time" goes to infinity, and thus classical theory of global (or uniform) attractors is not applicable. A pullback attractor is a parameterized family {A (t)} t∈R of nonempty compact sets of the state space, which attracts bounded deterministic sets starting from earlier time. In recent years the upper semicontinuity of pullback attractors for dynamical systems with different kind of perturbations has also been widely studied (see, e.g., [2,6,10,13,14,19,21,23,24]). Simply speaking, if {A ε (t)} t∈R is the pullback attractor generated by the perturbed dynamical systems and {A 0 (t)} t∈R is the pullback attractor for the unperturbed one, then we say that In this paper, we consider the upper semicontinuity of pullback attractors for the following strongly damped wave equation: , (u(r), ∂ t u(r)) = (u r , u r ), u(x, t)| ∂ ×[r,∞) = 0, (1.1) where ⊂ R 3 ia a bounded smooth domain, and ε ∈ [0, 1]. For the nonlinearity f ∈ C 2 (R) with f (0) = 0" we assume that it satisfies: where λ 1 > 0 is the first eigenvalue of -. The external force g(x, t) is assumed to satisfy: g(x, t), ∂ t g(x, t) ∈ L 2 loc (R; L 2 ( )), and t -∞ e σ s g(x, s) 2 ds < ∞ for all t ∈ R, (1.4) where the positive constant σ will be settled in the proof of Lemma 3.1.
The dynamic behavior of analogous equations have been analyzed in the literature under different hypotheses. In the autonomous case (i.e., the forcing term g(x, t) = g(x)), the wellposedness, existence, and regularity of global attractors have been studied extensively for more general damped wave equations [4,5,8,9,12,15,16], and the exponential attractors and dimension estimates for global attractors are considered in [11,15,17]. In the nonautonomous case, we refer the readers to [20][21][22]25] and references therein.
When ε = 0, Eq. (1.1) reduces to the usual wave equation without strong damping term -∂ t u. Our main purpose in this paper is to study the limiting behavior of Eq. (1.1) as ε goes to 0. More precisely, we will prove the upper semicontinuity of pullback attractors in H 1 0 ( ) × L 2 ( ) for Eq. (1.1), that is, that the pullback attractor {A ε (t)} t∈R (ε ∈ [0, 1]) for Eq. (1.1) satisfies For Eq. (1.1), if the initial data belong to H 1 0 ( ) × L 2 ( ), then its solution is always in H 1 0 ( ) × L 2 ( ) and has no higher regularity, and we cannot obtain the compactness property by showing the boundedness of solutions in higher regular phase spaces. In this paper, we apply the techniques of Zelik [25] to overcome this difficulty and establish the asymptotic compactness of solution operators with perturbations (see Lemmas 3.3-3.5).
The structure of the paper is as follows. In the next section, we first recall some basic concepts and conclusions of the theory of pullback attractors and then prove an abstract result for verifying the upper semicontinuity of pullback attractors (Theorem 2.2), by applying which we prove the upper semicontinuity of pullback attractors for Eq. (1.1) in Sect. 3.
We introduce some notation that will be used in the paper. We denote by ·, · and · the inner product and norm in L 2 ( ), respectively. Let H α = D((-) α scale of Hilbert spaces generated by the Laplacian with Dirichlet boundary conditions on L 2 ( ) (see [18] for more detail) and endowed with standard inner products and norms, respectively, In particular, Generalized Poincaré inequality: We define the product Hilbert spaces as follows: For any given function u(t), we shortly write Throughout the paper, the symbols C and Q stand for a generic positive constant and a generic positive increasing function, respectively.

Preliminaries
In this section, we collect some basic facts from general theory of pullback attractors (see, e.g., [1,3]) and then state an abstract result for verifying the upper semicontinuity of pullback attractors.
Let us define a nonautonomous dynamical system by a process on a Banach space X with norm · X , that is, a family of continuous mappings U(t, τ ) : X → X, t ≥ τ , such that U(τ , τ ) = Id and U(t, s)U(s, τ ) = U(t, τ ) for all t ≥ s ≥ τ .

Definition 2.1 A family of compact sets
Definition 2.2 A family of sets D = {D(t)} t∈R is said to be pullback absorbing with respect to U(·, ·) if for every t ∈ R and any bounded D ⊂ X, there exists T > 0 (which depends on t and D) such that

Definition 2.3
A process U(·, ·) is said to be pullback D -asymptotically compact in X if for any t ∈ R and any sequences τ n n→∞ − → ∞ and x n ∈ D(tτ n ), the sequence {U(t, tτ n )x n } n∈N is relatively compact in X. Theorem 2.1 (see [3]) Let a family D = {D(t)} t∈R be pullback absorbing, and let U(·, ·) be pullback D -asymptotically compact in X. Then the family A = {A(t)} t∈R defined by is a pullback attractor for U(·, ·). Moreover, if for any t ∈ R, there exists T > 0 (which depends on t) such that Proof From Theorem 2.1 we immediately get (i). Taking any sequence , without loss of generality, we let x n ∈ A ε n (t n ). Then we can find sequences τ n → ∞ and ξ n ∈ D ε n (aτ n ) such that Then from the assumptions we easily obtain the precompactness of {x n } nN .
The theory for verifying the upper semicontinuity of pullback attractors has been considered by many authors; see [7,13] and references therein. By applying the ideas of [13, Theorem 4.1] we get the following result. Theorem 2.2 Let X, Y be two Banach spaces with norms · X and · Y , respectively, and let X be continuously embedded into Y . Assume that for every ε ∈ [0, μ], a process U ε (·, ·) has a pullback absorbing family D ε = {D ε (t)} t∈R satisfying (2.2), and a pullback attractor A ε = {A ε (t)} t∈R is given by Theorem 2.1. Suppose the following assumptions hold: Step 1. We prove that under our assumptions, for any t ∈ R, any sequences {ε n } n∈N ⊂ [0, μ] such that ε n n→∞ − → ε 0 , and y n ∈ A ε n (t), there exist y 0 ∈ A ε 0 (t) and a sub- By assumption (i) we let and by assumption (iii), without loss of generality, for each m, we let Let y 0,m = U ε 0 (t, aτ m )z m , and, without loss of generality, let By assumption (ii) we find Then, combining (2.9) and (2.10), for any δ > 0, we can find m δ , N δ ∈ N such that From assumption (iii) we also know that {y n } n∈N is precompact in X. Thus by (2.11), noting that X → Y , we immediately get (2.7).
Step 2. We claim that for any sequences such that for any δ > 0, we can find N ∈ N large enough such that Let τ > 0. We can find y n ∈ A ε n (aτ ) such that By (2.7), without loss of generality, we let and using assumption (ii), we readily check that On the other hand, assumption (iii) implies that sequences {x n } n∈N and {x * n } n∈N are precompact in X, from which, together with (2.14), noting that X → Y , we immediately obtain (2.12). Now we are ready to prove (2.6). If not true, then we can find δ > 0, sequences It follows from (2.12) that we can extract a subsequence {x n k } k∈N from {x n } n∈N and x * k ∈ A ε 0 (t n k ) such that which contradicts (2.15). The proof is completed.

Upper semicontinuity of pullback attractors for Eq. (1.1)
The existence and uniqueness of (weak) solutions u to Eq. (1.1) is classical (see, e.g., [4,5]) and can be obtained by the standard Faedo-Galerkin method. Such solutions satisfy: for any ε ≥ 0 and [r, T] ⊂ R, As a consequence, for any ε ≥ 0, we can construct the process U ε (t, r) associated with Eq. (1.1) as follows: and the mapping U ε (t, r) : E → E is continuous.
The main result of this section can be stated as follows.
Taking the limits as n → ∞ yields This finishes the proof.
To obtain the regularity estimates, we will apply the ideas of Zelik [25]. Split the solution U(t, r)ξ u (r) = u(t) of Eq. (1.1) as follows: where V (t, r)ξ v (r) = ξ v (t) and W (t, r)ξ w (r) = ξ w (t) solve the following equations, respectively: where L > 0 will be settled in the proof of Lemma 3.3.
Finally, combining (3.65) and (3.68) and taking the H α+1 -norm of both sides of (3.67), we get the desired conclusion. The proof is finished.
The proof of Theorem 3.1 The existence of pullback attractors follows directly from Theorem 2.1, Corollary 3.1, and Lemmas 3.3 and 3.5. Then by Lemmas 2.1, 3.2, 3.3, and 3.5 we readily check that all assumptions of Theorem 2.2 (with X = E and Y = E -1 ) are satisfied.