Existences and upper semi-continuity of pullback attractors in $H^1(\mathbb{R}^N)$ for non-autonomous reaction-diffusion equations perturbed by multiplicative noise

Wenqiang Zhao

Published 2014-11-28Version 1

The existences and upper semi-continuity of $\mathcal{D}$-pullback attractors in $H^1(\mathbb{R}^N)$ are proved for stochastic reaction-diffusion equation on $\mathbb{R}^N$ driven by a multiplicative noise and a deterministic non-autonomous forcing. It is also showed that the upper semi-continuity of the obtained attractors may happen in $H^1(\mathbb{R}^N)$ at any $\varepsilon_0\neq0$. To solve this problem, some abstract results are given, which imply that a family of attractors obtained in \emph{a initial space} are compact, attracting and upper semi-continuous in\emph{ a associated non-initial space} only if some compactness conditions of the cocycles in this space are assumed.