Random dynamics of fractional stochastic reaction-diffusion equations on Rn without uniqueness

Abstract

This paper is concerned with the asymptotic behavior of the solutions of the nonautonomous fractional stochastic reaction-diffusion equations on Rn with continuous (but not necessarily differentiable) nonlinear drift terms, which leads to the nonuniqueness of solutions and hence a multivalued random dynamical system with the solution operators of the equation. We first show the existence and uniqueness of random attractors for such a dynamical system and then establish the upper semicontinuity of these attractors as the intensity of noise approaches zero. The measurability of the random attractors is proved by the method of weak upper semicontinuity of multivalued functions, and the pullback asymptotic compactness of the system is derived by the idea of uniform estimates on the tails of the solutions.