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 $\rho_\lambda$, $\lambda>0$, with thick tails whose second moment is not bounded. We show that these steady state solutions are stable, and find basins of attraction for them using an entropy functional ${\mathcal H}_\lambda$ 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 ${\mathcal H}_\lambda$. While the entropy dissipation for ${\mathcal H}_\lambda$ is strictly positive, it turns out to be a difference of two terms, neither of which need 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 $\rho_\lambda$. In the present paper, we do not provide any estimate of the rate of convergence, but we discuss how this would result from a stability result for a certain sharp Gagliardo-Nirenberg-Sobolev inequality.