Blowing Up Critical Exponent For A Class Of Semilinear Parabolic Systems

Parabolic differential equation responses to a number of physics, chemistry, biology, and has realized ample achievements [1] - [2]. Since 1966 Fujita H. began to study the non-linear equations, a number of mathematicians like Levine H A. Pinsky R G. Escobedo M. mathematicians have begun to pay attention to the study of parabolic equation of nonlinear, and they have made great achievements on good wave solutions, branches and stability of the positive equilibrium solution, and asymptotic nature of the explosion, which not only enriched the methods of mathematics study, but also provided basis to change the world[3] - [11].This paper considered semi-linear parabolic equations of the Cauchy problem:where s∈Z⁺,p_i＞0,m_i∈(-2,+∞),u_i⁰(x),i=1,2…,s is defined as non-negative continuous function in R^N.Where m＞-2,n＞-2,p_i≥0,q_i≥0,i=1,2,u₀(x),v₀(x) are definedas non-negative continuous function in R^N. The main results are as following:1,The Blowing up critical exponent of equation (â… ):(1)Suppose u₀(x)≥0, v₀(x)≥0 and u_i⁰(x)≠0, if 1＜γ＜1＋2/N(1＋β),i=1,2…,s,Then the solution of equaton (â… ) is nonglobal.(2)Suppose p_i＞1,γ＞1＋2/N(1＋β), Then the solutions ofequation (â… ) exist globally for u_i⁰(x) small enough and blowing-up in finite time for u_i⁰(x) large enough.Whereβ=(?),β_i=(?),i=1,2…,s.2,The Blowing up critical exponent of equation (2)(1)Suppose u₀(x)≥0, v₀(x)≥0 andδ≠0, if max{α,β}＞N/2,Thenthe solution of equation(â…¡) is nonglobal.(2)Suppose u₀(x)≥0,v₀(x)≥0 andδ≠0 and max{α,β}＜N/2,Then the solution of equation (â…¡) exist globally for u₀(x),v₀(x)small enough and p₁＋q₁＞1,blowing-up in finite time foru₀(x),v₀(x) large enough.Where A=(?),δ=det(A－I),ifδ≠0, let x=(α,β)^T be the unique solution of (?).