On the Blowup of Multidimensional Semilinear Heat Equations

This work is concerned with positive, blowing-up solutions of the semilinear heat equation u_t − ∆u = u^p in 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 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.