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 in the general framework of weak (or “free energy”) solutions associated to initial data with finite mass M<8π, 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 H¹ × H² exponential stability of the self-similar profile in the quasiparabolic–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.