Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski–Poisson system

Tomasz Cieślak; Philippe Laurençot

doi:10.1016/j.crma.2009.01.016

Partial Differential Equations

Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski–Poisson system
[Explosion en temps fini des solutions à symétrie radiale d'un système de Smoluchowski–Poisson quasilinéaire critique]

Présenté par : Haïm Brezis

Tomasz Cieślak ¹ ; Philippe Laurençot ²

¹ Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
² Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université de Toulouse, 118, route de Narbonne, 31062 Toulouse cedex 9, France

Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 237-242.

Résumé
Abstract

L'explosion en temps fini est établie pour des solutions à symétrie radiale d'un système de Smoluchowski–Poisson quasilinéaire critique dès que la masse de la donnée initiale dépasse un certain seuil. Dans le cas surcritique, l'explosion peut se produire pour toute masse positive. L'argument principal de la démonstration est une nouvelle identité de type viriel.

Finite time blow-up is shown to occur for radially symmetric solutions to a critical quasilinear Smoluchowski–Poisson system provided that the mass of the initial condition exceeds an explicit threshold. In the supercritical case, blow-up is shown to take place for any positive mass. The proof relies on a novel identity of virial type.

Reçu le : 2008-10-15
Accepté le : 2009-01-06
Publié le : 2009-02-10

DOI : 10.1016/j.crma.2009.01.016

Affiliations des auteurs :

Tomasz Cieślak ¹ ; Philippe Laurençot ²

¹ Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
² Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université de Toulouse, 118, route de Narbonne, 31062 Toulouse cedex 9, France

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     author = {Tomasz Cie\'slak and Philippe Lauren\c{c}ot},
     title = {Finite time blow-up for radially symmetric solutions to a critical quasilinear {Smoluchowski{\textendash}Poisson} system},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {237--242},
     publisher = {Elsevier},
     volume = {347},
     number = {5-6},
     year = {2009},
     doi = {10.1016/j.crma.2009.01.016},
     language = {en},
}

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Tomasz Cieślak; Philippe Laurençot. Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski–Poisson system. Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 237-242. doi : 10.1016/j.crma.2009.01.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.01.016/

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