On the Blowup of Multidimensional Semilinear Heat Equations

This work is concerned with positive, blowing-up solutions of the semilinear heat equation uₜ − ∆u = uᵖ in Rⁿ. 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: uₜ − ∆u = uᵖ dans Rⁿ. 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.