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 ϱλ, 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 ϱλ.