CONTINUITY OF SOLUTIONS IN REACTION-DIFFUSION EQUATIONS AND ITS APPLICATIONS TO PULLBACK ATTRACTOR

In this paper, we consider the continuity of solutions for nonautonomous stochastic reaction-diffusion equation driven by additive noise over a Wiener probability space. It is proved that the solutions are strongly 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 meaunder 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.