无界区域上具有确定性非自治项的随机强阻尼波动方程的渐近行为

Many systems in the nature are inevitably influenced by time-dependent forces and uncertain factors. How to accurately describe the evolution of these systems? Such problem can be abstracted to study the asymptotic behavior of stochastic differential equation with deterministic non-autonomous term. This research has important theoretical and application significance. This project is to investigate the asymptotic behavior of stochastic strongly damped wave equation with deterministic non-autonomous term, which is an important class of stochastic differential equation. The main purpose is to study the existence and continuity of the random attractor of the equation with white noise on unbounded domains. In order to overcome the difficulties caused by the deterministic non-autonomous term and the stochastic term, we will first introduce two dynamical systems. One of them is for the deterministic non-autonomous term and the other is an ergodic metric dynamical system corresponding to the stochastic term. We will then establish the random dynamical system (RDS) driven by these two dynamical systems. Finally, using solution-decomposition techniques and various estimations to overcome the difficulty caused by the unboundedness of the domain, we will prove the existence and upper semicontinuity of the random attractor. The results of this project will help us to understand the evolution of the RDSs generated by the equations and lay the theoretical and methodological foundations for studying related issues.

自然界中的许多系统难免会受到与时间有关的外力和不确定因素的影响，怎样才能更为精确的描述这些系统的演化规律？这类问题可以抽象为对具有确定性非自治项的随机微分方程的渐近行为进行研究，对此研究具有重要的理论和应用意义。本项目将研究一类重要的随机微分方程——具有确定性非自治项的随机强阻尼波动方程的渐近行为，主要探讨在无界区域上方程受到白噪声扰动情况下随机吸引子的存在性和连续性问题。针对确定性非自治项和随机项对解的影响不同，拟引进两个动力系统——对应于确定性非自治项的动力系统和对应于随机项的遍历度量动力系统，建立由这两个动力系统共同驱动的随机动力系统，然后利用解的分解技巧和估计方法克服由区域无界性引起的验证系统紧性的困难，最终证明随机吸引子的存在性和连续性。本项目的结果将有助于我们理解由方程生成的随机动力系统的演化规律，并为相关问题的研究奠定理论和方法基础。

本项目基本按照原计划完成，并主要取得了如下的一些结果：.（1）无界区域上非自治随机强阻尼波动方程的随机吸引子存在性；.（2）在无限序列的加权空间中连续过程存在拉回指数吸引子一些充分条件；.（3）具有时变耦合系数的二阶格点系统在无限序列的加权空间中拉回指数吸引子的存在性和连续性；.（4）无界区域上带有可加噪声的随机强阻尼波动方程的随机吸引子的存在性。