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

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 init...

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Veröffentlicht in:Communications on pure and applied analysis 2012, Vol.11 (1), p.47-60
Hauptverfasser: Blanchet, Adrien, Laurençot, Philippe
Format: Artikel
Sprache:eng
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Zusammenfassung: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 $d\geq 3$.
ISSN:1553-5258
0010-3640
1097-0312
DOI:10.3934/cpaa.2012.11.47