The Existence Of Solutions To Some Nonlocal Evolution Equations

The thesis mainly considers two classes of problems:abstract fractional integro-differential equations with nonlocal initial conditions; reaction-diffusion equation with nonlocal source. It is arranged as follows:Chapter 1 gives the background of the research on the thesis and introduces the main work of the thesis.Chapter 2 deal with the Cauchy problems of abstract fractional integro-differen-tial equations involving nonlocal initial conditions inα—norm, where the operator in the linear part is the generator of a compact analytic semigroup. New criterions, ensuring the existence of mild solutions, are established. The results are obtained by using the theory of operator families associated with the function of Wright type and the semigroup generated by the operator in the linear part, Krasnoselkii's fixed point theorem and Schauder's fixed point theorem.Chapter 3 considers the extinction properties of positive solutions for reaction-diffusion equation with nonlocal source where m,p,q∈(0,1), d,k,a>0,Ω(?)RN(N>2) is a bounded domain with a smooth boundary. It shows that m= p+q is a critical extinction exponent of positive solutions for the equation. Moreover, the precise decay estimates of solutions before the occurrence of the extinction are obtained.In Chapter 4, an application to a fractional partial integro-differential equation with nonlocal initial condition is also considered.