( L 2 , H 1 ) -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 ${ℝ}^n$ and prove the existence of an $(L^2,H^1)$-random attractor for such a random dynamical system. The nonlinearity $f$ is supposed to satisfy the growth of arbitrary order $p-1$ ($p\ge 2$). The $(L^2,H^1)$-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.