Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model

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.