Non-autonomous and random attractors for delay random semilinear equations without uniqueness

Abstract

We first prove the existence and uniqueness of pullback and random attractors for abstract multi-valued non-autonomous and random dynamical systems. The standard assumption of compactness of these systems can be replaced by the assumption of asymptotic compactness. Then, we apply the abstract theory to handle a random reaction-diffusion equation with memory or delay terms which can be considered on the complete past defined by R−.In particular, we do not assume the uniqueness of solutions of these equations.