Upper semicontinuity of pullback attractors for a nonautonomous damped wave equation

Abstract

AbstractIn 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 $\{A_{\varepsilon }(t)\}_{t\in \mathbb{R}}$ { A ε ( t ) } t ∈ R of Eq. (1.1) with $\varepsilon \in [0,1]$ ε ∈ [ 0 , 1 ] satisfies $\lim_{\varepsilon \to \varepsilon _{0}}\sup_{t\in [a,b]} \operatorname{dist}_{H_{0}^{1}\times L^{2}}(A_{\varepsilon }(t),A_{ \varepsilon _{0}}(t))=0$ lim ε → ε 0 sup t ∈ [ a , b ] dist H 0 1 × L 2 ( A ε ( t ) , A ε 0 ( t ) ) = 0 for any $[a,b]\subset \mathbb{R}$ [ a , b ] ⊂ R and $\varepsilon _{0}\in [0,1]$ ε 0 ∈ [ 0 , 1 ] .