Posedness Stochastic Delay Equations With Poisson Jumps Development, Stability And Overall Attractive Set And Control

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.