Existences and upper semi-continuity of pullback attractors in H^1(R^N) for non-autonomous reaction-diffusion equations perturbed by multiplicative noise - Wenqiang Zhao (Electronic journal of differential equations, 2016)
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.