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We investigate the long time behavior of the critical massPatlak-Keller-Segel equation. This equation has a one parameter family ofsteady-state solutions $\rho_\lambda$, $\lambda>0$, with thick tails whosesecond moment is not bounded. We show that these steady state solutions arestable, 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 anentropy-entropy dissipation inequality for ${\mathcal H}_\lambda$. While theentropy dissipation for ${\mathcal H}_\lambda$ is strictly positive, it turnsout to be a difference of two terms, neither of which need to be small when thedissipation is small. We introduce a strategy of "controlled concentration" todeal with this issue, and then use the regularity obtained from theentropy-entropy dissipation inequality to prove the existence of basins ofattraction for each stationary state composed by certain initial dataconverging towards $\rho_\lambda$. In the present paper, we do not provide anyestimate of the rate of convergence, but we discuss how this would result froma stability result for a certain sharp Gagliardo-Nirenberg-Sobolev inequality.

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