Finite degrees of freedom  for the refined blow-up profile  of the semilinear heat equation

Nguyen, Van Tien; Zaag, Hatem

doi:10.24033/asens.2344

Finite degrees of freedom for the refined blow-up profile of the semilinear heat equation
[Nombre fini de degrés de liberté du profil raffiné de l'équation semilinéaire de la chaleur]

Nguyen, Van Tien ; Zaag, Hatem

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 5, pp. 1241-1282.

Résumé
Abstract

Nous raffinons le comportement asymptotique des solutions de l'équation semilinéaire de la chaleur avec une non-linéarité sous-critique au sens de Sobolev, qui explosent en temps fini à un point d'explosion avec le profil communément admis comme générique. Pour obtenir ce raffinement, nous devons abandonner le profil explicite comme premier ordre de l'approximation, et prenons à la place une fonction non explicite comme première description du comportement au voisinage de la singularité. Cette fonction non explicite est en fait une solution spécifique que nous construisons, obéissant à un certain comportement prescrit. La construction repose sur la réduction du problème à un problème en dimension finie et l'utilisation d'un argument topologique basé sur la théorie du degré pour conclure. De façon étonnante, on constate que le nouveau profil non explicite produit une famille avec un nombre fini de degrés de liberté, soit $\frac{(N + 1) N}{2}$ si $N$ est la dimension de l'espace.

We refine the asymptotic behavior of solutions to the semilinear heat equation with Sobolev subcritical power nonlinearity which blow up in some finite time at a blow-up point where the (supposed to be generic) profile holds. In order to obtain this refinement, we have to abandon the explicit profile function as a first order approximation, and take a non explicit function as a first order description of the singular behavior. This non explicit function is in fact a special solution which we construct, obeying some refined prescribed behavior. The construction relies on the reduction of the problem to a finite dimensional one and the use of a topological argument based on index theory to conclude. Surprisingly, the new non explicit profiles which we construct make a family with finite degrees of freedom, namely $\frac{N (N + 1)}{2}$ if $N$ is the dimension of the space.

Publié le : 2017-11-12

DOI : 10.24033/asens.2344

Classification : 35K58, 35K55; 35B40, 35B44.
Keywords: Semilinear heat equations, finite-time blow-up, blow-up profile, stability.
Mot clés : Équation semilinéaire de la chaleur, explosion en temps fini, profil à l'explosion, stabilité.

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     author = {Nguyen, Van Tien and Zaag, Hatem},
     title = {Finite degrees of freedom  for the refined blow-up profile  of the semilinear heat equation},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1241--1282},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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Nguyen, Van Tien; Zaag, Hatem. Finite degrees of freedom  for the refined blow-up profile  of the semilinear heat equation. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 5, pp. 1241-1282. doi : 10.24033/asens.2344. http://www.numdam.org/articles/10.24033/asens.2344/

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