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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Blow-up rate of type II and the braid group theory
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by Noriko Mizoguchi PDF
Trans. Amer. Math. Soc. 363 (2011), 1419-1443 Request permission


A solution $u$ of a Cauchy problem or a Cauchy-Dirichlet problem for a semilinear heat equation \[ u_t = \Delta u + u^p \] with nonnegative initial data $u_0$ is said to undergo type II blow-up at $t = T$ if \[ \limsup _{t \nearrow T} \; (T-t)^{1/(p-1)} |u(t)|_\infty = \infty . \] 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:
  • Received by editor(s): July 2, 2007
  • Received by editor(s) in revised form: May 15, 2009
  • Published electronically: October 20, 2010
  • © Copyright 2010 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 363 (2011), 1419-1443
  • MSC (2000): Primary 35K20, 35K55
  • DOI:
  • MathSciNet review: 2737271