A sufficient condition for blowup solutions of nonlinear heat equations

AbstractThe 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