(L2,H1)-Random Attractors for Stochastic Reaction-Diffusion Equation on Unbounded Domains

Abstract

We study the random dynamical system generated by a stochastic reaction-diffusion equation with additive noise on the whole space ℝn and prove the existence of an (L2,H1)-random attractor for such a random dynamical system. The nonlinearity f is supposed to satisfy the growth of arbitrary order p-1 (p≥2). The (L2,H1)-asymptotic compactness of the random dynamical system is obtained by using an extended version of the tail estimate method introduced by Wang (1999) and the cut-off technique.