Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion

Adrien Blanchet; Philippe Laurençot

doi:10.3934/cpaa.2012.11.47

Article Contents

2012, Volume 11, Issue 1: 47-60. Doi: 10.3934/cpaa.2012.11.47

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Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion

Adrien Blanchet^1, and
Philippe Laurençot^2,

1.
GREMAQ, CNRS UMR 5604, INRA UMR 1291, Université de Toulouse, 21 Allée de Brienne, F--31000 Toulouse
2.
Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université de Toulouse, F–31062 Toulouse cedex 9

Received: November 2009

Revised: October 2010

Published: January 2012

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Abstract

For a specific choice of the diffusion, the parabolic-elliptic Patlak-Keller-Segel system with non-linear diffusion (also referred to as the quasi-linear Smoluchowski-Poisson equation) exhibits an interesting threshold phenomenon: there is a critical mass $M_c>0$ such that all solutions with initial data of mass smaller or equal to $M_c$ exist globally while the solution blows up in finite time for a large class of initial data with mass greater than $M_c$. Unlike in space dimension $2$, finite mass self-similar blowing-up solutions are shown to exist in space dimension $d\geq 3$.

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Mathematics Subject Classification: Primary: 35K65, 34C10; Secondary: 92C17.

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References

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