Upper semicontinuity of pullback attractors for damped wave equations

TL;DR

This paper proves the upper semicontinuity of pullback attractors for a damped wave equation, showing their stability under parameter variations and precompactness in the associated function space.

Contribution

It establishes the upper semicontinuity of pullback attractors for damped wave equations with respect to a parameter, under certain assumptions, and demonstrates their precompactness.

Findings

Pullback attractors depend continuously on parameters.

The union of attractors over parameters is precompact.

Results hold in the $H_0^1 \times L^2$ function space.

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 $H_0^1 (\Omega) \times L^2(\Omega)$.