A sufficient condition for blowup solutions of nonlinear heat equations

The author discusses the initial-boundary value problem (u(i))(1) = Deltau(i) + f(i) (u(1),..., u(m))with u(i)\partial derivativeOmega =0 and u(i)(x,0) =Phi(1)(x), i = 1,..., m, in a bounded domain Omega subset of R-11. Under suitable assumptions on fi, he proves that, if Phi(i) greater than or equal to, (1 + epsilon(0))Psi(i) in Di subset of Omega, for some small epsilon(0) > 0, then the solutions blow up in a finite time, where Psi(i) is a positive solution of DeltaPsi(i) + fi(Psi(I),..., Psi(m)) greater than or equal to 0, with Psi(i)\partial derivative(Di) = 0 for i = 1,..., m. If m = 1, the initial value can be negative in a subset of Omega. (C) 2004 Elsevier Inc. All rights reserved.