The critical exponents of parabolic equations and blow-up in R n

Source 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 > ( n − 2) + / n , p > max (1, m ) and σ > − 1 if n = 1 or σ > − 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 > 1, every nontrivial solution blows up in finite time. But for p > p c a positive global solution exists.