Containing The Blasting Of The Existence Of Solutions Of The P-laplace Heat Equation As A Whole Reconciliation

Partial Differential Equation (i.e. PDE) has developed for several hundred years, in which Heat Equation attracts many people's attention. It is invariable for people to study whether heat equations exist global solutions and when , where and how they will blow up. In this paper we consider the following heat, equation with p-Laplaceand establish the existence of global solution to the equation under some conditions and give two sufficient conditions for blowing up of local solution in finite time, where is a smooth bounded domain in , In 2001,Tan Zhong([1]) has solved the existence and asymptotic estimates of global solutions and finite time blowup of local solution of chemical reaction diffusion equation with special diffusion coefficient, i.e. the following formwhere 1) is a smooth bounded domain inThis equation has some chemical meaning and developed quickly in the past ten years. In this paper we will extend this problem to more general condition, i.e. the p-Laplacian condition. We will proof the existence of global solution by Hardy-Sobolev inequality and find two sufficient conditions of blowing up in finite time byvariational methods and classical concave methods ([2] [7] [11] [16]). The conclusions are following:Theorem 1.1 If u0(x) ∈Σ 1, then (1.1) exists a global solution u(x,t;uo). Moreover, there exists a a > 0, such that ||Vu(i)|| = O(e~at), t â†'∞.Theorem 1.2 Let u(x,t; u0) be the solution of (1.1) and E(UQ) < 0, then the solution u(x, t; u0) will blow up in finite time.Theorem 1.3 Let u(x,t; u0) be the solution of (1.1) and U0 ∈Σ2, then the solution u(x,t;u0) will blow up in finite time.