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

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 :
Accepté le :
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DOI : 10.1016/j.crma.2009.01.016

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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     title = {Finite time blow-up for radially symmetric solutions to a critical quasilinear {Smoluchowski{\textendash}Poisson} system},
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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/

[1] A. Blanchet, J.A. Carrillo, Ph. Laurençot, Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, in press

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[3] T. Cieślak, Ph. Laurençot, Looking for critical nonlinearity in the one-dimensional quasilinear Smoluchowski–Poisson system, in preparation

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