The Research Of Solution To The Non-local Diffusion Equation With The Non-linear Source Reaction

In the past few decades, the nonlinear diffusion as a kind of important partial differential equations is received extensive attention of scholars at home and abroad. The nonlinear diffusion equations involve the mathematical model in the field of physics, mechanics, biology and chemistry, etc., are used to describe the diffusion, heat transfer, population dynamics, the evolution of the epidemic in the communication, biological population and migration, etc., so they have a broad application background. The diffusion term of the classical diffusion equation reflects by Laplace operator, but the Laplace operator can only reflect the local case, such as objects of heat transfer by high temperature to low temperature, the spread of the material proceeds by the higher concentration to the lower one, the movement of the population migrates from the big density to the small one. In fact, the space of the nonlocal effect is widespread in the nature. For example, a species will move in the larger space scope while the diffusion motions will be associated with a probability distribution, which leads to the space of the nonlocal effects occur.This thesis studied the following the blow-up of the solution for a non-local diffusion problem with Neumann boundary conditions and a reaction term: and here α> 0, Ω is a bounded connected and smooth domain, the kernel J:RN â†' R is assumed to be a non-negative, bounded and symmetric continuous function J(x)= J(-x), ∫RNJ(x)dx= 1. The initial datum uo(x) is a non-negative, non-trivial and bounded continuous function. By using Banach’s fixed point theorem, we obtained the existence and uniqueness of solutions. Also we proved the solution of the equations above will blow up in finite time and got the blow-up rate by using the theory of differential Last we obtained the blow-up set of the first equation by using constructing functions and maximum principle.