Journal of Differential Equations 169, 588613 (2001) Further Study on a Nonlinear Heat Equation 1 Changfeng Gui Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada Wei-Ming Ni School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455 and Xuefeng Wang Department of Mathematics, Tulane University, New Orleans, Louisiana 70118 Received May 4, 1999 dedicated to professor jack hale on the occasion of his 70th birthday 1. INTRODUCTION In this paper, we continue our study in [GNW] and [Wa1] on the Cauchy problem { u t =2u +u p in R n _(0, T ), (1.1) u( x, 0)=.( x) in R n , where 2= n i =1 ( 2 x 2 i ) is the Laplace operator, p >1, T>0 and .0 is a given bounded continuous nonnegative function in R n . It is well-known that there exists T=T[.]>0 such that (1.1) has a unique classical solu- tion u( x, t; .) in C 2, 1 ( R n _(0, T )) & C( R n _[0, T )) which is bounded in R n _[0, T $] for all T $<T[ .], and &u(}, t; .)& L ( R n ) as t A T[.] if T[ .]<. We call u( x, t; .) the global solution to (1.1) if T[ .]=, and we say u( x, t; .) blows up in finite time if T[ .]<. Problem (1.1) has a simple appearance but rich mathematical structure. The task to answer all the questions turns out to be rather demanding. All doi:10.1006jdeq.2000.3909, available online at http:www.idealibrary.com on 588 0022-039601 35.00 Copyright 2001 by Academic Press All rights of reproduction in any form reserved. 1 This work is supported partially by NSF and NSERC. View metadata, citation and similar papers at core.ac.uk