Continuity of the blow-up profile with respect to initial data and to the blow-up point for a semilinear heat equation

Khenissy, S.; Rébaï, Y.; Zaag, H.

Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 1, pp. 1-26.

We consider blow-up solutions for semilinear heat equations with Sobolev subcritical power nonlinearity. Given a blow-up point $\hat{a}$, we have from earlier literature, the asymptotic behavior in similarity variables. Our aim is to discuss the stability of that behavior, with respect to perturbations in the blow-up point and in initial data. Introducing the notion of “profile order”, we show that it is upper semicontinuous, and continuous only at points where it is a local minimum.

Nous considérons des solutions explosives de l'équation semilinéaire de la chaleur avec une nonlinéarité sous-critique au sens de Sobolev. Etant donné un point d'explosion $\hat{a}$, grâce à des travaux antérieurs, on connaît le comportement asymptotique des solutions en variables auto-similaires. Notre objectif est de discuter la stabilité de ce comportement, par rapport à des perturbations du point d'explosion et de la donnée initiale. Introduisant la notion de « l'ordre du profil », nous montrons qu'il est semi-continu supérieurement, et continu uniquement aux points où il est un minimum local.

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