Approximation Solutions For Some Linear Nonlocal Dispersal Equations

The nonlocal dispersal equation has been widely used to model different dis-persal processes,such as in material science,population dynamics,epidemiology.Recently,much attention has been paid to the study of nonlocal dispersal equa-tions.This thesis is concerned with the following nonlocal dispersal equationut(x,t)=?RN G(x,y)u(y,t)dy-u(x,t),x?RN,t>0.here G is a nonnegative dispersal kernel function.We study the existence and uniqueness of the solutions to the above equation with Dirichlet and Neumann boundary conditions.We then consider the asymptotic behavior of the solution when the dispersal kernel is rescaled.The main results include the following two parts.Firstly,we study the nonlocal Dirichlet problem when the kernel function G(x,y)is J(x,x-y)and J(y,x-y),respectively,where J is a non-negative integrable function.We establish the global existence and uniqueness of solutions to the above equation by using Banach fixed point theorem and the comparison principle when the initial value is given.And obtain the solutions of classical heat equation can be approximated by the solutions of linear nonlocal diffusion equation when the kernel is J(x,x-y).On the other hand,we prove that the solutions of linear nonlocal dispersal equations with rascaled kernel function J(y,x-y)can convergence to the solutions of convection-diffusion equation.Next,we study the solutions of nonlocal dispersal problem with Neumann boundary condition,and give the existence and uniqueness of the solutions when the kernel function is J(x,x-y).We show that the solutions of Neumann problem for the heat equation can be approximat,ed by the solutions of nonlocal dispersal equation with the rescaled kernel functions.