Pull Back Attractor For Stochastic Semilinear Strongly Damped Wave Equations In Locally Uniform Spaces

In this paper,we consider the Cauchy problem in Rn for the stochastic strongly damped semilinear wave equations with additive noise.The nonlinear term f grows like |u|q with 0<q<(n + 2)/(n-2),Wj are independent two-sided real-valued Winer process.In recent years,the research field of stochastic dynamical systems has attracted more and more attention of scholars have been in-depth research and rapid development in theory and application.Long-time behavior for analogous equations on bounded domain have been investigated by many authors.The existence of the attractor and the structure of the attractor characteristics have been proved on bounded domain in many papers.For the unbounded domain,due to the Sobolev embedding is no longer compact,Sobolev space nesting formula is no longer valid,and the classical Sobolev space does not include traveling wave solutions and constant solutions,etc.Therefore,the general Sobolev space as the phase space of the above equation is still not ideal.For related problems,a number of authors have proved the existence of the attractor of equations by thinking about weighted spaces and bounded and uniformly continuous function spaces,as well as the correlation property of the locally uniform space.Due to the stochastic strongly damped wave equation has an infinite propagation speed of initial disturbances,the traditional proof method of strong asymptotic tightness can not be used directly applied in proving the existence of the attractor.In this paper,the compactness properties are proved by using the compactness property of weak forms.In this paper,it is prove the global solvability and the existence of the pull back attractor to above problem in the locally uniform space X =Wlu2,p(Rn)× Llup(Rn).Special attention in this paper is devoted to the asymptotic compactness of the corresponding semigroup S(t,?)since the equations has no strong compactness condition in the phase space.To overcome this difficulty we first prove the sel B1:= S(1,?)?+(B0)is bounded in D(L)=Wlu2,p(Rn)×Wlu2,p(Rn),where B0 is the absorbing set of S(t,?)in X,then we use the compact embedding theorems Wlu2,p(Rn)× Wlu2,p(Rn)(?)W?1,p(Rn)× W?1,p(Rn)obtain the compactness of the set B1 in the phase space X.