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

• Published in: Electronic journal of differential equations Vol. 2016; no. 294; pp. 1 - 28
• Main Author: Wenqiang Zhao
• Format: Journal Article
• Language: English
• Published: Texas State University 16.11.2016
• Subjects: non-autonomous reaction-diffusion equation; pullback attractor; Random dynamical systems; upper semi-continuity
• ISSN: 1072-6691

In this article, we establish sufficient conditions on the existence and upper semi-continuity of pullback attractors in some non-initial spaces for non-autonomous random dynamical systems. As an application, we prove the existence and upper semi-continuity of pullback attractors in $H^1(\mathbb{R}^N)$ are proved for stochastic non-autonomous reaction-diffusion equation driven by a Wiener type multiplicative noise as well as a non-autonomous forcing. The asymptotic compactness of solutions in $H^1(\mathbb{R}^N)$ is proved by the well-known tail estimate technique and the estimate of the integral of $L^{2p-2}$-norm of truncation of solutions over a compact interval.