# American Institute of Mathematical Sciences

April  2002, 8(2): 435-450. doi: 10.3934/dcds.2002.8.435

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

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

Revised  October 2001 Published  January 2002

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 $\partial_t u=\Delta u + |u|^{p-1}u$.

Citation: Frank Merle, Hatem Zaag. O.D.E. type behavior of blow-up solutions of nonlinear heat equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 435-450. doi: 10.3934/dcds.2002.8.435
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