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 $\{A_{\epsilon }(t){\}}_{t\in \mathbb{R}}$} of Eq.(1.1) with $\epsilon \in [0,1]$ satisfies that for any $[a,b]\subset \mathbb{R}$ and ${\epsilon }_0\in [0,1]$, ${\lim }_{\epsilon \to {\epsilon }_0}{\sup }_{t\in [a,b]}{\mathrm{dist}}_{H_0^1\times L^2}(A_{\epsilon }(t),A_{{\epsilon }_0}(t))=0$, and ${\cup }_{t\in [a,b]}{\cup }_{\epsilon \in [0,1]}{A}_{\epsilon }(t)$ is precompact in $H_0^1(\Omega )\times L^2(\Omega )$.

arXiv commentThis paper has been withdrawn by the author due to a crucial sign error in p11,p13