Research On Wong-Zakai Approximation And Regularity Of Non-autonomous Stochastic P-laplacian Equation

In this paper,we investigate a non-autonomous stochastic quasi-linear parabolic equation driven by multiplicative white noise on the entire space RN,1≤N ∈N by a Wong-Zakai approximation technique:du=(div(|▽u|p-2▽u)-λu+f(t,x,u)+g(t,x))dt+u(?)dW(t),t>τ,x∈RN,where g∈Lloc2(R,L2(RN)),λ>0 and p>2 are constants,f is a nonlinear function satisfying some dissipative conditions,W(t)is two-sided real valued Brownian motions on a probability space,W(t,ω)is abbreviated as ω(t),(?)signifies the Stratonovich sense of the stochastic term.Defining Gδ(vtω)=(ω(t+δ)-ω(t))/δ,it is easy to check that Gδ(vtw)is a stochastic processes with stationary increment,than Gδ(vtω)may be regarded as an approximation of the white noise that for every T>0,see[18].For arbitrary t ∈ R,ω ∈ Ω,let Wδ=Wδ(t,ω)=∫0tGδ(vrω)dr.In this paper,we study the following point-wise deterministic quasi-linear parabolic equations driven by the process Wδ:duδ=(div(|▽uδ|p-2▽uδ)-λuδ+f(t,x,uδ)+g(t,x))dt+uδdWδ.Note that the above is a random non-autonomous differential equation and its solutions admit a non-autonomous random dynamical system.In this paper,we consider the following three research work:Fristly,in terms of the convergence property of Brownian motions,we show that the limit of solutions of deterministic differential equations in L2(RN)is a solution of stochastic differential equation,which is equivalent to the following It? stochastic differential equations:du=(div(|▽u|p-2▽u)-λu+1/2u+f(t,x,u)+g(t,x))dt+udW(t);see Theorem 3.9.The upper semi-continuity of their random attractors in L2(RN)is proved in Theorem 3.11.Our second work in this paper is to establish the Wong-Zakai approximation in the Banach space Lq(q≥p>2),where q is the growth exponent of the nonlinearity.To this end,some further compactness in Lq is needed.This is achieved by a truncation approach,by which we prove that the solution vanishes in Lq(q≥p>2)on a domain on which the solution of deterministic differential equations diverges to positive and negative infinite.Then by the theorem in[36],we obtain that the random attractor of deterministic differential equations converges to that of stochastic p-Laplacian equations in Lq(RN)(q≥p>2)in the sense of upper semi-continuity;see Theorem 4.4.Finally,with some small additional assumptions on the coefficients,by using an induction technique we obtain the solution is bounded in Ll(RN)for arbitrary l>q.By Sobolev interpolation,we establish some asymptotical compactness of random attractors in Ll(RN).Following this,we obtain the Wong-Zakai approximations in this higher-regular space Ll(RN)for arbitrary l>q;see Theorem 5.4.