Wong-Zakai Approximation And Regularity For The Stochastic Reaction-diffusion Equations

The Wong-Zakai approximation was proposed by Wong and Zakai in 1965.Its main idea is to consider the approximation problem of stochastic differential equations driven by white noise by approximating the Brownian motion by a smooth stochastic processes.There have been significant achievements regarding the existence of attractors for stochastic partial differential equations driven by stationary processes and WongZakai approximation in the initial space.However,the solutions of stochastic partial differential equations possess certain regularity,so considering the Wong-Zakai approximation problem in high order spaces has important theoretical values.When the nonlinear function satisfies certain monotonicity property,the Wong-Zakai approximation problem for stochastic reaction-diffusion equations in high regularity spaces has also been well studied.In this paper,we study the following non-autonomous stochastic reaction-diffusion equation with an additive white noise:du=(Δu-λu)dt+f(t,x,u)dt+g(t,x)dt+h(x)dW(t),t>τ,x ∈RN,(1)where λ>0,g∈Lloc2(R,L2(RN)),h∈H2(RN)∩ W2,p(RN),h(x)dW(t)is white noise,and W(t)=W(t,ω)=ω(t)is a Brownian motion over a probability space,andΔ is the Laplacian,and f is the nonlinear function with the growth exponent p.For the Wong-Zakai approximation equation of(1),we introduce the following path-wise deterministic equation driven by a stationary process:duδ=(Δuδ-λuδ)dt+f(t,x,uδ)dt+g(t,x)dt+h(x)gδ(vtω)dt,(2)where vt is the Wiener transform on the probability space,gδ(vtω)=ω(t+δ)-ω(t)/δ t∈R,δ≠0.Firstly,the article introduces a new method called nonlinear function decomposition.With this method,it is proved the solutions of equation(2)converge to the solutions of equation(1)in the L2(RN)space.As a result,the upper semi-continuous convergence of the random attractors of equation(2)to the random attractor of equation(1)is established in the L2(RN)space.Please see Theorem 3.5 and Theorem 3.6 for more details.Secondly,with this technique the convergence of solutions of the approximate equation to the solutions of the stochastic equation is demonstrated in Lp(RN)space by calculations.By combining this result with the upper semi-continuity conclusion in the L2(RN)space,the upper semi-continuity of the random attractors in the Lp(RN)space is established.Please see Theorem 4.4 for more details.Finally,the boundedness of the solutions of the approximate equation and the stochastic equation is proved in the space Ll(RN)(?l>p)using mathematical induction.And by Sobolev interpolation inequalities,the difference inequality of the solutions is established.Ultimately,the upper semi-continuity of the random attractors is established in the Ll(RN)(?l>p)space.Please see Theorem 5.4 for more details.