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The present paper deals with the parabolic-parabolic Keller-Segel equation inthe plane inthe general framework of weak (or "free energy") solutionsassociated to an initial datum with finite mass $M\textless{} 8\pi$, finitesecond log-moment and finite entropy. The aim of the paper is twofold:(1) Weprove the uniqueness of the "free energy" solution. The proof uses aDiPerna-Lions renormalizing argument which makes possible to get the "optimalregularity" as well as an estimate of the difference of two possible solutionsin the critical $L^{4/3}$ Lebesgue norm similarly as for the $2d$ vorticityNavier-Stokes equation. (2) We prove a radially symmetric and polynomialweighted $L^2$ exponential stability of the self-similar profile in the quasiparabolic-elliptic regime. The proof is based on a perturbation argument whichtakes advantage of the exponential stability of the self-similar profile forthe parabolic-elliptic Keller-Segel equation established by Campos-Dolbeaultand Egana-Mischler.

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