Uniqueness and long time asymptotics for the parabolic-parabolic Keller-Segel equation

The present paper deals with the parabolic-parabolic Keller-Segel equation in the plane inthe general framework of weak (or "free energy") solutions associated to an initial datum with finite mass $M\textless{} 8\pi$, finite second log-moment and finite entropy. The aim of the paper is twofold:(1) We prove the uniqueness of the "free energy" solution. The proof uses a DiPerna-Lions renormalizing argument which makes possible to get the "optimal regularity" as well as an estimate of the difference of two possible solutions in the critical $L^{4/3}$ Lebesgue norm similarly as for the $2d$ vorticity Navier-Stokes equation. (2) We prove a radially symmetric and polynomial weighted $L^2$ exponential stability of the self-similar profile in the quasi parabolic-elliptic regime. The proof is based on a perturbation argument which takes advantage of the exponential stability of the self-similar profile for the parabolic-elliptic Keller-Segel equation established by Campos-Dolbeault and Egana-Mischler.