The Existence Of Pullback Attractor For A Viscoelastic Stochastic Wave Equation With Critical Exponents On Unbounded Domains

We study the asymptotic behavior for the following damped semilinear stochastic wave equation defined on the unbounded domain Rn(n≤3) with a linear memory tern and additive noise.Where, α and λ are positive numbers, the intergral is a memory term, f is a nonlinear function, g is a external force, the last one is a stochastic noise and Wis an independent tow-sided real-valued Winer process. In this paper, we assume that, where n≤3, the nonlinearity has an arbitrary growth for n=1,2and is cubic growth rate for n=3. It is well known that the Poincare’embeddings are not compact on the undounded domain. This lead to prove the existence of the attractors is difficult. It is more difficult that prove the existence of the attractors for the random wave equation on undounded domain. we prove the existence of the attractors for the wave equation with memory term. We put the asymptotic compactness of the prove is divided into two part for solve that the Poincare’embeddings are not compact. In the first part, the first step we use the uniform estimates method to prove the solution Χ=(u(t,τ,ω,u0),v(t,τ,ω,v0),η(t,τ,ω,η0)) of wave equation on Rn; the second step we derive estimates on solution of wave equation in Rn\Qk and Qk. In the second part, first, we prove the existence of pullback attracting set for the above-mentioned wave equation on H(Rn)×L2(Rn)×M2(Rn). Then, the asymptotic compactness of the above-mentioned wave equation corresponding random dynamical system is proved by using the energy equation method. And hence we prove the existence of pullback attractor for above-mentioned wave equation.