A sufficient condition for blowup solutions of nonlinear heat equations ✩

The author discusses the initial-boundary value problem (ui )t = Δui + fi(u1, . . . , um) with ui |∂Ω = 0 and ui(x, 0) = φi (x), i = 1, . . . , m, in a bounded domain Ω ⊂ Rn. Under suitable assumptions on fi , he proves that, if φi (1 + ε0)ψi in Di ⊂ Ω, for some small ε0 > 0, then the solutions blow up in a finite time, where ψi is a positive solution of Δψi + fi(ψ1, . . . , ψm) 0, with ψi |∂Di = 0 for i = 1, . . . , m. If m = 1, the initial value can be negative in a subset of Ω.  2004 Elsevier Inc. All rights reserved