The Blow-up Solutions To A Class Of Reaction-diffusion Equation With Initial And Boundary Conditions

In this paper, we consider the blow-up solutions to a class of reaction-diffusion equation with initial and boundary conditionsut- A?u = up,u(x, t)|Dc= 0,u(x, 0) = u0(x).where x ? D, D =(0, 1), Dc= R1\D.A?u(x) = C?P.V.?D?D?u(y)- u(x)|x- y|1+?dy.where P.V. stands for Cauchy principal value and C?=2?-1??((1+?)/2)?1/2?(1-?/2).In chapter one, we give the development and application of reaction-diffusion equations, including physics, biology and Earth sciences neighborhood, the sources of nonlocal Fractional Laplacian operator, research methods of blowing up, and the status of the domestic and foreign research results.In chapter two, we give some definitions of nonlocal calculus, theorems and formulas,and list some estimations of semigroup and some embedding theorems of Sobolev space.In chapter three, we give two kinds of blowing up. That is point blows up and norm blows up. We obtain the local existence of the solution about time in the appropriate space by fixed point theorem and the theorems of analytical semigroup theory when p > 1. For point blows up, In nonlocal reaction diffusion equation is given maximum principle, and then prove the basis of comparison principles, we construct the blow-up lower solution by original equation's exact solution, then we can know the existence of blow-up solution by comparison principle and obtain the blow-up times and blow-up rates. For norm blows up,the blow-up criterion for this kind of equation is studied by using the analytic semigroup theory and continuation theorem. Applying this criterion and an estimate based on nonenergy positive energy initial conditions Lp+1?blow-up solutions, the dependence of p on the growth index of the reaction term and the non local index ? were estimated. Specifically:When 0 < ? < 1, p ?(1,11-?], When 1 ? ? < 2, p > 1 is arbitrary, the solution blows up in the sense of Lp+1norm, From the phenomenon of the classical diffusion and abnormal(nonlocal) diffusion phenomenon.In chapter four, we list some problems and possible methods to solve them.