Journal
JOURNAL OF FUNCTIONAL ANALYSIS
Volume 262, Issue 5, Pages 2142-2230Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jfa.2011.12.012
Keywords
Keller-Segel model; Critical mass; Basins of attraction; Gradient flows with respect to transport distances
Categories
Funding
- project EVaMEF [ANR-09-JCJC-0096-01]
- U.S. National Science Foundation [DMS 0901632]
- DGI-MCI (Spain)
- AGAUR-Generalitat de Catalunya [2009-SGR-345]
- [MTM2011-27739-004-02]
- ICREA Funding Source: Custom
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [0901632] Funding Source: National Science Foundation
- Agence Nationale de la Recherche (ANR) [ANR-09-JCJC-0096] Funding Source: Agence Nationale de la Recherche (ANR)
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We investigate the long time behavior of the critical mass Patlak-Keller-Segel equation. This equation has a one parameter family of steady-state solutions Q(lambda), lambda > 0, with thick tails whose second moment is unbounded. We show that these steady-state solutions are stable, and find basins of attraction for them using an entropy functional H-lambda, coming from the critical fast diffusion equation in R-2. We construct solutions of Patlak-Keller-Segel equation satisfying an entropy entropy dissipation inequality for H-lambda. While the entropy dissipation for H-lambda is strictly positive, it turns out to be a difference of two terms, neither of which needs to be small when the dissipation is small. We introduce a strategy of controlled concentration to deal with this issue, and then use the regularity obtained from the entropy-entropy dissipation inequality to prove the existence of basins of attraction for each stationary state composed by certain initial data converging towards Q(lambda). (C) 2011 Elsevier Inc. All rights reserved.
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