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

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- by Noriko Mizoguchi PDF
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**363**(2011), 1419-1443 Request permission

## Abstract:

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.## References

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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
- 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: https://doi.org/10.1090/S0002-9947-2010-04784-1
- MathSciNet review: 2737271