随机半线性强衰减波动方程在局部一致空间上的吸引子

摘要

在无界区域Rn中考虑了具有可加噪声的随机强衰减半线性波动方程的Cauchy问题,在相空间 X =W2,plu(Rn)× Lplu(Rn)中证明了该方程的整体可解性和随机吸引子的存在性.为解决该方程相关联的半群f(t,ω)的弱渐近紧性问题,首先证明了集合B1:= f(1,ω)γ+(B0)在空间D(L)= W 2,plu(Rn)×W 2,plu(Rn)中的有界性,其中B0是半群f(t,ω)在相空间 X中的吸收集;然后利用紧嵌入定理W 2,plu(Rn)× W 2,plu(Rn)?W 1,pρ(Rn)× W 1,pρ(Rn)得到了集合B1在相空间 X中的弱渐近紧性.%In this paper,we consider the Cauchy problem for the stochastic strongly damped semilinear wave equations with additive noisein the unbounded domain-Rn.The global solvability and the existence of the sto-chastic attractor to this problem are proved in the phase space X = W2,plu(Rn)× Lplu(Rn).To study to the asymptotic compactness of the corresponding semigroup f(t,ω),we first prove the set B1:= f(1,ω)γ+(B0) is bounded in D(L)= W2,plu(Rn)×W2,plu(Rn),where B0 is the absorbing set of in f(t,ω)in X,then we use the compact embedding theorems W2,plu(Rn)×W2,plu(Rn)? W1,pρ(Rn)×W1,pρ(Rn)obtain the compactness of the set B1 in the phase space X.