Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35K20, 35K55

Retrieve articles in all journals with MSC (2000): 35K20, 35K55

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

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

American Mathematical Society