Pullback Attractors For Multi-valued Mean-square Random Dynamical Systems Generated By Stochastic Parabolic Equations With Infinite Delays

In this article, we study the asymptotic behavior of solutions of a stochastic parabolic equation with infinite delays and nonlinear multiplicative noise. We first present the related concepts and a sufficient and necessary condition for the existence of pullback attractors of multi-valued mean-square random dynamical systems. Sec-ondly, we do not assume any Lipschitz condition on the nonlinear term, just a conti-nuity assumption together with growth conditions so that the solutions of the Cauchy problem exist, but the uniqueness fails to be true. Then we introduce the definition of multi-valued mean-square random dynamical systems. Finally, using the theory of multi-valued mean-square random dynamical systems and a new method for checking the asymptotical upper-semicompactness of the solutions, we prove the existence and uniqueness of a compact pullback attractor in the sense of mean-square topology.