The critical exponents of parabolic equations and blow-up in<i>Rⁿ</i>

Abstract

In this paper we study the Cauchy problem in R n of general parabolic equations which take the form u t = Δ u m + t s | x | σ u p with non-negative initial value. Here s ≧ 0, m &gt; ( n − 2) + / n , p &gt; max (1, m ) and σ &gt; − 1 if n = 1 or σ &gt; − 2 if n ≧ 2. We prove, among other things, that for p ≦ p c , where p c ≡ m + s ( m − 1) + (2 + 2 s + σ)/ n &gt; 1, every nontrivial solution blows up in finite time. But for p &gt; p c a positive global solution exists.