Upper semicontinuity of pullback attractors for damped wave equations

Abstract

In this paper, we study the upper semicontinuity of pullback attractors for a strongly damped wave equation. In particular, under some proper assumptions, we prove that, the pullback attractor $\{A_\varepsilon(t)\}_{t\in\mathbb R}$} of Eq.(1.1) with $\varepsilon\in[0,1]$ satisfies that for any $[a,b]\subset\mathbb R$ and $\varepsilon_0\in[0,1]$, $\lim_{\varepsilon\to\varepsilon_0} \sup_{t\in[a,b]} \mathrm{dist}_{H_0^1\times L^2} (A_\varepsilon(t), A_{\varepsilon_0}(t))=0$, and $\cup_{t\in[a,b]} \cup_{\varepsilon\in[0,1]} A_\varepsilon(t)$ is precompact in $H_0^1 (\Omega) \times L^2(\Omega)$.