Regularity of random attractors for stochastic reaction-diffusion equations on unbounded domains

The existence of a unique random attractors in H-1(R-n) for a stochastic reaction-diffusion equation with time-dependent external forces is proved. Due to the presence of both random and non-autonomous deterministic terms, we use a new theory of random attractors which is introduced in [B. Wang, J. Differential Equations 253 (2012) 1544-1583] instead of the usual one. The asymptotic compactness of solutions in H-1(R-n) is established by combining "tail estimate" technique and some new estimates on solutions. This work improves some recent results about the regularity of random attractors for stochastic reaction-diffusion equations.