Global Asymptotic Stability Of Two Kinds Of Non-local Evolution Equations In Bounded Domain

Non-local evolution equations as a very important part of the differential equa-tions, in the research of living habits and features in the real world, have very important significance.This thesis studies two kinds of non-local evolution equations in bounded domain: one is the differential equation with delay and non-local effects, the species is divided into two classes with age:the matured and the unmatured, considering the case that the unmatured move while the matured don’t; the other one is the reaction-diffusion equation with delay and non-local effects, the population is also divided into two age classes with a fixed maturation, considering the case that both the unmatured and the matured move.This thesis is formed by the following four chapters:In the first chapter, we describe the background and development of the non-local evolution equations. and illustrate the main work, the method and significance of the context.In the second chapter, we introduce some basic concepts and dynamic properties of attracting intervals, and present some basic notations and definitions.In the third chapter, we mainly consider the case that the unmatured move but the matured don’t, study the differential equation with non-local effects and time delays. we get existence and uniqueness of the equilibrium of this equation by analyzing and discussing the corresponding abstract integral equation. We use the methods of attracting intervals to research, obtaining the global asymptotic stability of the equilibrium and the optimal criteria that can judge the global asymptotic stability of this non-local evolution equation.In the fourth chapter, we mainly consider the case that both the matured and the unmatured move, study the non-local reaction diffusion equation. We use the knowledge of partial differential theory to study the corresponding abstract integral equation to get the existence and uniqueness of the equilibrium of the non-local delayed reaction diffusion equation. we prove global asymptotic stability of this equation and get the optimal criteria that can judge global asymptotic stability of the equilibrium of equations by using the idea of attracting intervals to studying the equilibrium of the equation. Finally, some examples are giving.