非自治多值随机动力系统及其在波方程中的应用

The theory of pullback attractors for multi-valued non-compact random dynamical systems and a method of proving asymptotic compactness based on the concepts of the Kuratowski measure of the non-compactness of a bounded set are used to establish the existence of pullback attractors for the multi-valued non-compact random dynamical systems. Due to the lack of continuity of set-valued mappings, we have to discuss the measurability of the attractor. Stochastic wave equation is a kind of important partial differential equation, it usually describes all kinds of wave phenomenon in the nature. The difficulty is cause by the complicated structure of the solution, the specific form and growth index of damping term and a nonlinear term in wave equation. It is hard to establish the upper semicontinuity of the random attractors, when the wave equation become a parabolic equation under some disturbance and the structure of the solution will be changed. We will be studied the applications of the stochastic wave equation under the framework of the non-autonomous multivalued random dynamical system, we are interesting in the existence、regularity、measurability and the upper semicontinuity of the attractor. It is useful to clarify the nature and the asymptotic behavior of the solution for the stochastic wave equations..Razumikhin method will be used in the study of mean-squared random dynamic systems to get the ultimate boundedness and convergence properties in the sense of mean square. The defined of mean-square random attractor describe the long time behavior of solutions, the mean square attractor is a weakly compact random collection set in the topology of the mean-square.

在非紧性多值随机动力系统拉回吸引子的理论中, 以非紧性测度为基础的渐近紧性方法, 用来证明拉回吸引子的存在性。由于集值映射连续性的缺失，吸引子可测性的研究成为必须。随机波方程是一类重要的偏微分方程，它通常描述自然界中各种各样的波动现象。解的结构复杂性增加了讨论难度，同时方程中阻尼项和非线性项的具体形式及增长指数也会给研究带来困难。当波方程扰动为抛物方程时，解的结构发生变化，如何讨论吸引子上半连续性也是一个重点。本项目组将比较系统地研究随机波方程在非自治多值随机动力系统理论框架下的应用，从拉回吸引子的存在性、正则性、可测性及上半连续性等方面进行研究，这些研究对阐明随机波方程解的性质和渐近性行为很有意义。.均方随机动力系统的讨论过程中利用 Razumikhin 方法一致估计解的有界性并建立均方意义下解的收敛性，均方随机吸引子用来描述解的长时间行为，是一个均方拓扑意义下的弱紧随机集合。

具有时滞项的波方程用于描述一些具有滞后效应的震荡现象的模型。本课题主要研究了无界域上具有线性记忆项和非线性阻尼项的随机变时滞随机波方程随机吸引子的存在性和唯一性，由于无界域上Sobolev紧嵌入的缺失，我们需要引入新的方法一致估计无界域部分并验证有界域部分的渐进紧性，由于方程中包含时滞项、非自治项、随机项，这使得方程生成的动力系统不仅是非自治的而且是随机的，我们将在非紧性随机动力系统的理论框架下证明吸引子存在性。本项目还研究了有界域上具有无穷时滞和非线性阻尼项的随机波方程，由于非线性项不满足Lipschitz连续，导致方程的解不唯一，这给解的可测性和拉回吸引子的可测性验证中带来新的困难，另外变时滞项和无穷时滞项会使紧性分析的过程更加复杂。在多值随机动力系统的理论框架下，我们得到了拉回吸引子的存在性和可测性。