  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.
