摘要
本论文的目的是研究几类连续的和不连续的乘性噪声驱动的随机时滞发展方程,如带泊松跳与时滞的随机偏微分方程的适定性、稳定性、整体吸引集、可控性以及一大类在Hilbert空间框架下随机时滞发展方程的一些细微的性质.本文主要使用的方法包括：Picard迭代法、半群方法(或温和解逼近)、随机分析的技术、不动点定理和积分不等式方法.本文的结构如下：
第一章给出论文一些背景介绍和预备知识.在1.1节中,介绍了随机时滞发展方程并给出了本文主要研究的问题.在1.2节中,给出了论文中将要用到的一些预备知识,包括随机It6积分、带泊松跳随机积分、对H>1/2的分数次布朗运动的Wiener积分、微积分方程、时滞微分方程、脉冲微分方程和一些有用的不等式.
第二章利用Picard迭代方法、半群方法和预解算子的理论得到了带泊松跳与时滞的中立型随机微分积分方程的适定性.在2.1节中,利用Picard迭代法和预解算子的理论,在一类广义Lipschitz条件下得到了带泊松跳与有限时滞的中立型随机微分积分方程温和解的存在唯一性.此外,利用Bihari不等式的一个推论给出了解对初始值的连续依赖性.为了说明在2.1节中结果的有效性,我们提供了热方程的一个例子.在2.2节中,同样利用2.1节中所用的方法得到了一类带泊松跳与无限时滞的中立型随机微分积分方程的适定性.另外,利用停时技术,在局部非Lipschitz条件下得到了此类方程温和解的存在唯一性.
第三章讨论了随机时滞发展方程的整体吸引集.我们的方法是基于不动点方法和使用一些适当的积分不等式.在3.1节中,通过建立两个新的脉冲积分不等式分别得到了带泊松跳的脉冲中立型随机偏函数微分积分方程温和解的整体吸引集与拟不变集.此外,我们还得出一些充分条件,以确保在矩指数稳定性和几乎必然指数稳定性的意义下温和解的稳定性.在3.2节中,利用Banach不动点定理结合微积分方程的预解算子的理论考虑了一类对于H>1/2的分数次布朗运动中立型随机偏微分积分方程温和解的的适定性.此外,通过使用在4.1节中得到的积分不等式,我们得到了此方程中的一类方程的整体吸引集.
第四章考虑了一类带泊松跳与无限时滞的二阶脉冲中立型随机微分方程的可控性.利用有界线性算子的强连续余弦族理论、随机分析和Banach不动点定理得到了对带泊松跳与无限时滞的非局部二阶脉冲中立型随机函数微分方程的可控性一组新的充分条件.特别地,我们将此结论应用于带泊松跳与无限时滞的随机非线性波动方程.
第五章研究了于Hilbert空间中带泊松跳与有限时滞的随机二维Navier-Stokes方程在均方和几乎必然意义下弱解的渐近行为.
The objective of this dissertation is to study for some classes of stochastic delay evolution equations driven both by continuous and discontinuous multiplicative noise. Or in other words, more detailed, the inspiration of this dissertation is some charac-teristics of solutions, such as the well-posedness, stability, global attracting set and controllability for the stochastic partial differential equations with delay and Poisson jumps and some fine properties for a large class of stochastic delay evolution equa-tions in the Hilbert spaces framework. Main methods will be used in the disserta-tion include:Picard approximation method, the semigroup method (or mild solution approach), the techniques of stochastic analysis, fixed point theorems and integral inequalities method. Structure of this dissertation is as follows:
In Chapter1, the introduction and the preliminaries are given. In Section1.1, we introduce on the stochastic delay evolution equations as well as all problems will be studied in this dissertation. In Section1.2, we provide some preliminaries which may be applied to the remaining chapters throughout the dissertation, including stochastic integrals with Brownian motion, stochastic integrals with Poisson jumps, the Wiener integral for the fractional Brownian motion with H>1/2, integro-differential equa-tions, differential equations with delay, impulsive differential equations, and some useful inequalities.
In Chapter2, the well-posedness of neutral stochastic integro-differential equa-tions with delay and Poisson jumps is proved by the Picard type method of approxi-mation, the semigroup method and theory of resolvent operators. In Section2.1, under a class of generalized Lipschitz conditions, by using the method of Picard approxima-tion and the theory of resolvent operators, the existence and uniqueness of mild solu-tions for neutral stochastic integro-differential equations with finite delay and Poisson jumps is investigated. In addition, utilizing a corollary of Bihari's inequality, we ob-tain the stability through the continuous dependence of mild solutions on the initial data. As applications, to illustrate the effectiveness of the results achieved in Section2.1, we provide an example on heat equation. In Section2.2, also by the method used in Section2.1, we study the well-posedness for a class of neutral impulsive stochastic integro-differential equations with infinite delay and Poisson jumps. Moreover, the existence and uniqueness of mild solutions under local non-Lipschitz conditions is also given by means of the stopping time technique.
In Chapter3, we discuss the global attracting set for neutral stochastic delay evo-lution equations. Our approaches are based on a fixed point method and the method by using some appropriate integral inequalities. In Section3.1, by establishing two new impulsive-integral inequalities, the global attracting and quasi-invariant sets of the mild solution for neutral impulsive stochastic partial functional differential equa-tions with Poisson jumps are obtained, respectively. Moreover, we shall derive some sufficient conditions to ensure stability of this mild solution in the sense of both mo-ment exponential stability and almost surely exponential stability. In Section3.2, by using Banach fixed point theorem combined with theories of resolvent operators for integro-differential equations, the well-posedness of mild solutions for a class of neu-tral stochastic partial integro-differential equations driven by a fractional Brownian motion with Hurst index H€(1/2,1) is investigated. In addition, by using an inte-gral inequality as in Section3.1, we obtain the global attracting set for a class of this equations.
In Chapter4, the controllability of a class of second order neutral impulsive stochastic differential equations with infinite delay and Poisson jumps is considered. By using the theory of strongly continuous cosine families of bounded linear opera-tors, stochastic analysis techniques and with the help of the Banach fixed point the-orem, we derive a new set of sufficient conditions for the controllability of nonlocal second order neutral impulsive stochastic functional differential equations with infi-nite delay and Poisson jumps. Especially, an application to the stochastic nonlinear wave equation with infinite delay and Poisson jumps is given.
In Chapter5, we study the asymptotic behavior for the weak solutions of stochas-tic2D Navier-Stokes equations with finite memory and Poisson jumps in both mean square and almost sure senses by viewing the classical form of the stochastic Navier-Stoke equation as a semilinear stochastic evolution equation in Hilbert spaces.
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