The blow-up rate for a coupled system of semilinear heat equations with nonlinear boundary conditions

The paper deals with the blow-up rate of positive solutions of the system u(t) = u(xx) + u(l11)v(l12), v(t) = v(xx) + u(l21)v(l22) With nonlinear boundary conditions u(x)(0, t) = 0, u(x)(1, t) = (u(p11)v(p12))(1, t), and v(x)(0, t) = 0, v(x)(1, t) = (u(p21)v(p22))(1, t). Under some assumptions on the matrices L = (l(ij)) and P = (p(ij)) and on the initial data u(0), v(0), the solution (u, v) blows up at finite time T, and we prove that max(x is an element of[0,1]) u(x, t) (respectively, max(x is an element of[0,1]) v(x, t)) goes to infinity like (T - t)beta(1/2) (respectively, (T - t)(beta 2/2)) as t --> T, where beta(i) < 0 are the solutions of (L - Id)(beta(1), beta(2))(t) = (-1, -1)(t). (C) 1999 Elsevier Science Ltd. All rights reserved.