Tempered random attractors for parabolic equations in weighted spaces

This paper deals with the asymptotic behavior of solutions of parabolic equations on unbounded domains which contain stochastic terms as well as non-autonomous deterministic terms. We define a continuous random dynamical system for the equations in weighted Sobolev spaces that allow functions to have certain polynomial growth rate at infinity and hence include all bounded solutions. We prove pullback asymptotic compactness of solutions as well as the existence and uniqueness of tempered random attractors in the weighted spaces. The structures of the tempered attractors are fully characterized by tempered complete solutions. In the case where the non-autonomous deterministic terms are periodic in time, we show that the tempered random attractors are also periodic in time. To overcome the difficulty of non-compactness of Sobolev embeddings on unbounded domains, the idea of uniform pathwise estimates on the tails of solutions is employed to show asymptotic compactness.