Study On The Solvability And Stability For Several Kinds Of Evolution Equations

Evolution equations are widely used in physics,control theory,economic math-ematics,communication theory and biological mathematics.An important method to study the evolution equation is to transform the general equation into an abstract differ-ential evolution equation,and discuss the behavior of the solution of the system by using the theory and method in functional analysis.The fixed point theorem is an important tool to study the existence and uniqueness of mild solutions of evolution equations.The stability of solutions is also an important dynamic behavior in the theory of differential equations.In this paper,the existence,uniqueness and stability of mild solutions for several kinds of evolution equations are studied by using the theories and methods of fractional power operator,linear evolution system,successive approximation method,fixed point theorem,resolvent operator theory and stochastic convolution.The specific content consists of the following four parts.The chapter 1 is the introduction,which introduces the research background and present situation of evolution equations at home and abroad,summarizes the main work of this paper,and gives the definitions,lemmas and basic properties needed for the follow-up work.In chapter 2,we mainly study the existence and uniqueness of mild solution for nonautonomous neutral stochastic delay evolution equation in Hilbert space.Firstly,we discuss the well-posedness and continuity of the defined operatorΦby using frac-tional power operator,and then the sequence{(5⁹)}₉≥0)is Cauchy sequence according to the successive approximation method.Finally,the existence and uniqueness of mild solution are discussed by using the fixed point theorem.Because the unbounded linear operator family（）can generate a unique linear evolution system in Hilbert space,and the nonlinear term in the equation satisfies weaker conditions,the equation discussed in this chapter is more practical as a mathematical model.In chapter 3,the existence and regularity of solutions for a class of neutral stochas-tic partial functional integro-differential equations are researched in Hilbert space.Con-sidering the influence of practical factors,we add random terms to the existing research.First,we get the expression of mild solution by using resolvent operator theory,and then by utilizing Schauder fixed point theorem,we obtain the existence and uniqueness of mild solution in Hilbert space(3 and(3.In addition,it is verified that the mild solution of the equation is its classical solution under certain conditions.In chapter 4,we mainly study the existence and asymptotic stability in the-th moment of mild solution for a class of impulsive neutral stochastic partial functional integro-differential equations with variable delays and Poisson jumps.In order to wide-ly apply the studied problems to scientific fields,we consider the influence of Poisson jumps on practical phenomena.Under the assumption that the linear part of the equa-tion generates resolvent operators,the expression of mild solution for the equation is obtained by using the resolvent operator theory.Then,Combined with Banach fixed point theorem,we attain the asymptotic stability in the-th moment of mild solution,and the continuous dependence of mild solution on initial value is further discussed.