Looking for critical nonlinearity in the one-dimensional quasilinear Smoluchowski-Poisson system

Tomasz Cieślak; Philippe Laurençot

doi:10.3934/dcds.2010.26.417

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2010, Volume 26, Issue 2: 417-430. Doi: 10.3934/dcds.2010.26.417

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Looking for critical nonlinearity in the one-dimensional quasilinear Smoluchowski-Poisson system

Tomasz Cieślak^1, and
Philippe Laurençot^2,

1.
Institute of Applied Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa
2.
Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université de Toulouse, F–31062 Toulouse cedex 9

Received: January 31, 2009

Revised: April 30, 2009

Published: June 2010

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Abstract

It is known that classical solutions to the one-dimensional quasilinear Smoluchowski-Poisson system with nonlinear diffusion $a(u)=(1+u)^{-p}$ may blow up in finite time if $p>1$ and exist globally if $p<1$. The case $p=1$ thus appears to be critical but it turns out that all solutions are global also in that case. Two classes of diffusion coefficients are actually identified in this paper, one for which all solutions to the corresponding quasilinear Smoluchowski-Poisson system are global and the other one leading to finite time blowup for sufficiently concentrated initial data. The cornerstone of the proof are an alternative formulation of the Smoluchowski-Poisson system which relies on a novel change of variables and a virial identity.

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Mathematics Subject Classification: Primary: 35B45, 35K55; Secondary: 35B60, 35B33, 35B05.

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