Pullback Attractor For Stochastic Reaction-Diffusion Equations On Unbounded Domains

This paper is to study the following time-dependent stochastic reaction-diffusion equations with multiplicative noise on unbounded domainsFirst of all, we prove the existence and uniqueness of solutions of time-dependent stochastic reaction-diffusion equations with multiplicative noise. In order to get the solutions, we have to turn the stochastic equation into a deterministic one with random parameters. Given t∈R and ω∈Ω, let z(t,ωco)=e-aw(t), let v be a new variable given by v(t,τ,ω,vτ)=z(t,ω)u(t,τ,ω,uτ), withvτ=z(τ,ω)u. Then get the equal equation and the existence and uniqueness of solutions.Then, we will estimate the solution for the above equation and prove the pullback asymptotic compactness of the solution. It is difficult to consider the pullback asymptotic compactness of the solutions on unbounded domains, for the Sobolev embeddings are not compact on unbounded domains, to solve this problem, we recommend the idea of energy equations as introduced by Ball to improve the pullback asymptotic compactness of solutions. After that, we show that the cocycle associated with the equation has a measurable pullback absorbing set. Finally, we prove the existence of pullback attractors and characterize the structures of the pullback attractors.