A sufficient condition for blowup solutions of nonlinear heat equations

Abstract

The author discusses the initial-boundary value problem ( u i ) t = Δu i + f i ( u 1,…, u m ) with u i| ∂Ω=0 and u i ( x,0)= φ i ( x), i=1,…, m, in a bounded domain Ω⊂R n . Under suitable assumptions on f i , he proves that, if φ i ⩾(1+ ε 0) ψ i in D i⊂Ω , for some small ε 0>0, then the solutions blow up in a finite time, where ψ i is a positive solution of Δψ i + f i ( ψ 1,…, ψ m )⩾0, with ψ i | ∂D i =0 for i=1,…, m. If m=1, the initial value can be negative in a subset of Ω.