Backward Compactness And Almost Continuity Of Pullback Attractors For Non-autonomous Stochastic Kuramoto-Sivashinsky Lattice Equations

In this paper,we study the existence of backward compact random pullback attractors for non-autonomous random Kuramoto-Sivashinsky lattice equations and explore the almost continuity of random pullback attractors(residual dense continuity and full pre-continuity).This paper is divided into four parts.The first part introduces the research background and development status of dynamical system and attractor,as well as the background and research status of Kuramoto-Sivashinsky equation,and also gives a brief description of the references.The second part mainly introduces the theoretical knowledge used in this article.In the third part,we study the existence of backward compact random attractors in the Kuramoto-Sivashinsky lattice equation.Consider the following equation where Z denotes the interger set,i∈Z,v>0,λ>8 and β>2,g=(gi(t))∈l2 is non-autonomous external force term,and W(t)is a two-sided Wiener process defined on a probability space(Ω,F,P).In this part,it can be proved that the equation generates a stochastic dynamic system by making assumptions about the non-autonomous term in the equation and making it satisfy certain tempered conditions.Secondly,it can be proved that the dynamic system has a backward pullback absorption set through a series of proofs,and it can be proved that the dynamic system is backward asymptotically compact by using tail estimation.Finally,according to the existence and uniqueness theorem of the attractor,it can be proved that the dynamic system is the existential uniqueness of the backward compact pullback attractor.Finally,we study the almost continuity(residual dense continuity and full pre-continuity)of the Kuramoto-Sivashinsky lattice equations.Residual continuity means that all points on all residual subsets of the binary map(τ,s)→A(τ,θwω)on R2 are continuous.According to the article of Professor Li Yanrong,the important theoretical results have been established for the almost continuous pullback attractor.In order to prove the almost continuity of the pullback attractor,the local uniform compactness of the pull-back attractor and the joint continuity of the dynamical system are obtained by verifying the union closedness of the attracted universe on the time sample.Therefore,in this paper,we first prove the union closedness of the attracted universe on the time sample,then make further assumptions on the non-autonomous term by proving the existence of local absorption set and local asymptotic compactness,and finally prove that the dynamic system is joint continuous on the time,sample and initial data,so as to obtain the almost continuity of the pullback attractor.