Critical Exponents for the Blowup of Solutions with Sign Changes in a Semilinear Parabolic Equation, II

✍ Scribed by Noriko Mizoguchi; Eiji Yanagida

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 353 KB
• Volume: 145
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1997.3387

✦ Synopsis

The blowup of solutions of the Cauchy problem { u t =u xx + |u| p&1 u u(x, 0)=u 0 (x) in R\_(0, ), in R is studied. Let 4 k be the set of functions on R which change sign k times. It is shown that for p k =1+2Â(k+1), k=0, 1, 2, ... , any solution with u 0 # 4 k blows up in finite time if 1

p k . This is an extension of our previous result [17], in which a fast decay condition was imposed on initial data. It is also shown in this paper that if u 0 decays more slowly than |x| &2Â( p&1) as |x| Ä + , then the solution blows up in finite time regardless of the number of sign changes.

1998 Academic Press

Mathematics Subject Classification (1991): 35K15, 35K5.

1. Introduction

Since the pioneering work of Fujita [4], critical exponents for the blowup of solutions of nonlinear parabolic problems have been studied by many authors (see a survey paper of Levine [13] for detailed information on this subject). However, they are mainly dealing with positive solutions, and there are few results in the case where solutions may change sign.
In this paper we are concerned with the Cauchy problem
where p>1. We say that a solution blows up in finite time if the supremum norm of the solution diverges to as t Ä T for some T< .