Random attractors of reaction-diffusion equations without uniqueness driven by nonlinear colored noise

Abstract

This paper is concerned with the asymptotic behavior of solutions of the non-autonomous reaction-diffusion equations driven by nonlinear colored noise defined on unbounded domains. The nonlinear drift and diffusion terms are assumed to be continuous but not necessarily Lipschitz continuous which leads to the non-uniqueness of solutions. We prove the existence and uniqueness of pullback random attractors for the multi-valued non-autonomous cocycles generated by the solution operators. The measurability of the random attractors is established by the method based on the weak upper semicontinuity of the solutions. The asymptotic compactness of the solutions is derived by Ball’s idea of energy equations in order to overcome the non-compactness of Sobolev embeddings on unbounded domains.