Abstract
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 ρ{variant} λ, λ>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 Hλ coming from the critical fast diffusion equation in R2. We construct solutions of Patlak-Keller-Segel equation satisfying an entropy-entropy dissipation inequality for Hλ. While the entropy dissipation for Hλ 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 ρ{variant} λ.
Original language | English (US) |
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Pages (from-to) | 2142-2230 |
Number of pages | 89 |
Journal | Journal of Functional Analysis |
Volume | 262 |
Issue number | 5 |
DOIs | |
State | Published - Mar 1 2012 |
All Science Journal Classification (ASJC) codes
- Analysis
Keywords
- Basins of attraction
- Critical mass
- Gradient flows with respect to transport distances
- Keller-Segel model