无界域上带非线性阻尼和强阻尼随机波动方程解的性态

The Behavior of Solutions to Stochastic Wave Equations with Nonlinear Damping and Strong Damping on Unbounded Domains

本文研究无界域上带非线性阻尼、强阻尼以及可加噪声的非自治随机波动方程随机吸引子的存在性。首先证明该方程组的解可以定义一个随机动力系统,然后对方程的解进行一致估计得到此随机动力系统D-拉回随机吸收集的存在性,最后利用空间分割的方法克服无界域上Sobolev嵌入缺乏紧性的困难并证得此随机动力系统的D-拉回渐近紧性,进而得到该动力系统随机吸引子的存在性。

This article investigated the existence of random attractor for non-autonomous stochastic wave equation with nonlinear damping term and strong damping term, as well as additive noise on the unbounded domain. Firstly, it was proved that the solution of the equation can generate a random dynamical system, and then the uniform estimation of the solution was given to get the existence of D-pullback random absorbing set of the random dynamical system. Finally, the method of space division was used to overcome the non-compactness of Sobolev embedding on the unbounded domain and get the D-pullback asymptotical compactness of the random dynamical system, and then the existence of the random attractor of the dynamic system was obtained.