Abstract
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 {A_{varepsilon }(t)}_{tin mathbb{R}} of Eq. (1.1) with varepsilon in [0,1] satisfies lim_{varepsilon to varepsilon _{0}}sup_{tin [a,b]} operatorname{dist}_{H_{0}^{1}times L^{2}}(A_{varepsilon }(t),A_{ varepsilon _{0}}(t))=0 for any [a,b]subset mathbb{R} and varepsilon _{0}in [0,1].
Highlights
The theory of pullback attractors is a useful tool to study the long-time behavior of nonautonomous dynamical systems, in which the trajectory can be unbounded as “time” goes to infinity, and classical theory of global attractors is not applicable
We study the local uniformly upper semicontinuity of pullback attractors for a strongly damped wave equation
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
Summary
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 classical theory of global (or uniform) attractors is not applicable. We consider the upper semicontinuity of pullback attractors for the following strongly damped wave equation: 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
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