O.D.E. type behavior of blow-up solutions of nonlinear heat equations

Frank Merle; Hatem Zaag

doi:10.3934/dcds.2002.8.435

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2002, Volume 8, Issue 2: 435-450. Doi: 10.3934/dcds.2002.8.435

This issue Previous Article The problem Of blow-up in nonlinear parabolic equations Next Article Nonlinear parabolic differential equations and inequalities

O.D.E. type behavior of blow-up solutions of nonlinear heat equations

Frank Merle^1, and
Hatem Zaag^2,

1.
Département de Mathématiques, Université de Cergy-Pontoise and IUF, 2 Ave. Adolphe Chauvin, BP 222, Pontoise, 95 302 Cergy-Pontoise cedex
2.
Département de Mathématiques et Applications, CNRS École Normale Supérieure, 45 rue d'Ulm, 75 230 Paris cedex 05

Revised: October 2001

Published: April 2002

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Abstract

We show that all global (in time and in space) and bounded solutions of the vector-valued equation

$\frac{\partial w}{\partial s}=\Delta w-\frac{1}{2}y \cdot \nabla w -\frac{w}{p-1}+|w|^{p-1}w$

(where $w : \mathbb R^N\times \mathbb R \to \mathbb R^M, p> 1$ and $(N - 2)p < N + 2$) are independent of space and completely explicit. We then derive from this various uniform estimates and a uniform localization property for blow-up solutions of $\partial_t u=\Delta u + |u|^{p-1}u$.

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Mathematics Subject Classification: 35K40, 35K55, 35A20.

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