Upper semicontinuity of attractors of stochastic delay reaction-diffusion equations in the delay

Abstract

A system of stochastic delayed reaction-diffusion equations with multiplicative noise and deterministic non-autonomous forcing is considered. We first prove the existence and uniqueness of a bi-spatial pullback attractor for these equations when the initial space is C−ρ,0,L2O and the terminate space is C−ρ,0,H01O. The asymptotic compactness of solutions in C−ρ,0,H01O is established by combining “positive and negative truncations” technique and some new estimates on solutions. Then the periodicity of the random attractors is proved when the stochastic delay equations are forced by periodic functions. Finally, upper semicontinuity of the global random attractors in the delay is obtained as the length of time delay approaches zero.