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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Blow-up rate of type II and the braid group theory


Author: Noriko Mizoguchi
Journal: Trans. Amer. Math. Soc. 363 (2011), 1419-1443
MSC (2000): Primary 35K20, 35K55
Published electronically: October 20, 2010
MathSciNet review: 2737271
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Abstract: A solution $ u $ of a Cauchy problem or a Cauchy-Dirichlet problem for a semilinear heat equation

$\displaystyle u_t = \Delta u + u^p $

with nonnegative initial data $ u_0 $ is said to undergo type II blow-up at $ t = T $ if

$\displaystyle \limsup_{t \nearrow T} \; (T-t)^{1/(p-1)} \vert u(t)\vert _\infty = \infty. $

Let $ \varphi_\infty $ be the radially symmetric singular steady state of the Cauchy problem. Suppose that $ u_0 \in L^\infty $ is a radially symmetric function such that $ u_0 - \varphi_\infty $ and $ (u_0)_t $ change sign at most finitely many times. By application of the braid group theory, we determine the exact blow-up rate of solution with initial data $ u_0 $ which undergoes type II blow-up in the case of $ p > p_{_{JL}} $, where $ p_{_{JL}} $ is the exponent of Joseph and Lundgren.


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Additional Information

Noriko Mizoguchi
Affiliation: Department of Mathematics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan – and – Precursory Research for Embryonic Science and Technology, Japan Science and Technology Agency, 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan
Email: mizoguti@u-gakugei.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-04784-1
PII: S 0002-9947(2010)04784-1
Received by editor(s): July 2, 2007
Received by editor(s) in revised form: May 15, 2009
Published electronically: October 20, 2010
Article copyright: © Copyright 2010 American Mathematical Society