Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws

We consider a system of heat equations u(t) = Delta u and upsilon(t) = Delta upsilon in Omega x (0, T) completely coupled by nonlinear boundary conditions partial derivative u/partial derivative eta = e(p upsilon) u(alpha), partial derivative upsilon/partial derivative eta = u(g) e(beta upsilon) on partial derivative Omega x (0, T). We prove that the solutions always blow up in finite time for non-zero and non-negative initial values. Also, the blow-up only occurs on partial derivative Omega with C1(T - t) - p-beta/2(pq+beta-alpha beta) <= max u (x, t) <= C-2 (T-t) - p-beta/2(pq+beta-alpha beta) log (C-3(T - t) - q+1-alpha/2(pq+beta-alpha beta)) <= log (C-4(T-t) - q+1-alpha/2(pq+beta-alpha beta)) for p, q > 0, 0 <= alpha < 1 and 0 <= beta < p. (C) 2007 Elsevier Ltd. All rights reserved.