On the Blowup of Multidimensional Semilinear Heat Equations

Abstract

This work is concerned with positive, blowing-up solutions of the semilinear heat equation $u_t-∆u=u^p$ in ${\mathbf{R}}^n$. No symmetry assumptions are made. Working with the equation in similarity variables, we first prove a result suggested by center manifold theory. We then calculate the refined asymptotics for $u$ in a backward space-time parabola near a blowup point, and we obtain some information about the local structure of the blowup set. Our results suggest that in space dimension $n$, among solutions that follow the center manifold, there are exactly $n$ different blowup patterns.

Résumé

On étudie les solutions positives explosant en temps fini de l’equation semilinéaire de la chaleur : $u_t-∆u=u^p$ dans ${\mathbf{R}}^n$. On ne suppose aucune hypothèse de symétrie. On calcule le comportement asymptotique de la solution au voisinage d’un point d’explosion et on obtient certaines informations sur l’ensemble des points d’explosion.