JOURNAL OF DIFFERENTIAL EQUATIONS 77, 1O4-122(1989) The Blow-Up Rate of Solutions of Semilinear Heat Equations WENXIONG LIU * Department of Mathematics, Purdue University, West Lafayette, Indiana 47907 Received May 14, 1987; revised March 8, 1988 1. INTRODUCTION The purpose of this paper is to characterize the asymptotic behavior of the solution of the semilinear heat equation, u,-Au=e” in 52 x (0, T) u= -K on LX2 x (0, T) (1) u(x, 0) = q(x) > -K for XEQ, K>O, in a neighborhood of a blow-up point as t approaches the finite blow-up time T< co. The system (1) arises in a model for ignition (see [2]). Giga and Kohn [7,8,9] recently characterized the asymptotic behavior of the solution of u,-Au= ItdIP-l# in B x (0, T) u=o on LX2 x (0, T) (2) 4% 0) = cp(x) for XEQ near a blow-up point, provided n < 2, or n 2 3 and 1 <p < (n + 2)/(n - 2). They proved that (T- q%(x, t) + p, as t-+T,j?=l/(p-l), (3) uniformly for 1 x - a 1 < C( T- t) “* for some C, where a is any blow-up point. If n S 2, then (3) is true for any p < co. This fact suggestsa similar result * This work is partially supported by National Science Foundation Grant DMS-8420896. Present address: School of Mathematics, University of Minnesota, Minneapolis, MN 55455. 104 OO22-0396189 53.00 Copyright 0 1989 by Academtc Press, Inc All rights of reproduction m any form reserkd