Pullback Attractors For Reaction-diffusion Delay Equations On Unbounded Domains With Non-autonomous Deterministic And Stochastic Forcing Terms

This paper is concerned with the existence and regularity of pullback attrac-tors for non-compact random dynamical systems generated by reaction-diffusion delay equations with non-autonomous deterministic as well as stochastic forcing terms defined on an unbounded domain. The existence of a pullback random at-tractor in a space of higher regularity is established for the random dynamical system associated with the stochastic reaction-diffusion delay equation with addi-tive noise defined on all n-dimensional space, where the asymptotic compactnes for the random dynamical system with delays is demonstrated by using uniform a priori estimates for far-field values of solutions and a new method. Firstly, under some suitable conditions, we prove the existence of pullback absorbing sets for Eq.(1.1) by using the uniform estimates for the solutions. It is worth mentioning that we also show the measurability of pullback absorbing sets. Then we need to establish the existence of pullback attractors in CV for the equation on Rn. Since the domain Rn is unbounded and Sobolev embeddings are not compact in this case, we have to decompose the domain Rn into bounded and unbounded parts. Here we need to consider the interesting estimates of solutions on the complement of the large ball, and further decomposition of the equation defined in the large ball into finite and infinite dimensional parts. For the infinite dimensional part, we appeal to the idea of uniform estimates on the existence of absorbing sets to show that the norm of solutions are less than ε when ιis large enough. Finally, by using Arzela-Ascoli theorem, we prove that the solutions are uniformly bounded and equi-continuous in the finite dimensional case, where the uniform bounded-ness follows immediately from the uniform estimates of the solutions.