The critical exponents of parabolic equations and blow-up inRn

Abstract

In this paper we study the Cauchy problem inRnof general parabolic equations which take the formut= Δum+ts|x|σupwith non-negative initial value. Heres≧ 0,m> (n− 2)+/n,p> max (1,m) and σ > − 1 ifn= 1 or σ > − 2 ifn≧ 2. We prove, among other things, that forp≦pc, wherepc≡m+s(m− 1) + (2 + 2s+ σ)/n> 1, every nontrivial solution blows up in finite time. But forp>pca positive global solution exists.