随机半线性强衰减波动方程在局部一致空间上的吸引子

Attractor for stochastic semilinear strongly damped wave equations in locally uniform spaces

在无界区域R^n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=W_(lu)^(2,p)(R^n)×L_(lu)~p(R^n)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S(t,ω)的弱渐近紧性问题,首先证明了集合B_1∶=S(1,ω)γ^+(B_0)在空间D(L)=W_(lu)^(2,p)(R^n)×W_(lu)^(2,p)(R^n)中的有界性,其中B_0是半群S(t,ω)在相空间X中的吸收集;然后利用紧嵌入定理W_(lu)^(2,p)(R^n)×W_(lu)^(2,p)(R^n)■W_ρ^(1,p)(R^n)×W_ρ^(1,p)(R^n)得到了集合B_1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.