CONTINUITY OF SOLUTIONS IN <inline-formula><tex-math id="M1">$ H^1( {\mathbb{R}}^N)\cap L^{p}( {\mathbb{R}}^N) $</tex-math></inline-formula> FOR STOCHASTIC REACTION-DIFFUSION EQUATIONS AND ITS APPLICATIONS TO PULLBACK ATTRACTOR

Abstract

In this paper, we consider the continuity of solutions for non-autonomous stochastic reaction-diffusion equation driven by additive noise over a Wiener probability space. It is proved that the solutions are strongly continuous in $ H^1( {\mathbb{R}}^N)\cap L^p( {\mathbb{R}}^N) $ with respect to the $ L^2 $-initial data and the samples in the double limit sense. As applications of the results on the continuity we obtain that the pullback random attractor for this equation is measurable, compact and attracting in the topology of the space $ H^1( {\mathbb{R}}^N)\cap L^p( {\mathbb{R}}^N) $ under a weak assumption on the forcing term and the noise coefficient. More precisely, the continuity of solutions in the initial data implies the asymptotic compactness of system and therefore the attraction of attractor, and the continuity in the samples indicates its measurability. The main technique employed here is the difference estimate method, by which an appropriate multiplier is carefully selected.