  We consider the parabolic-parabolic Keller-Segel equation in the plane and
prove the nonlinear exponential stability of the self-similar profile in a
quasi parabolic-elliptic regime. We first perform a perturbation argument in
order to obtain exponential stability for the semigroup associated to part of
the first component of the linearized operator, by exploiting the exponential
stability of the linearized operator for the parabolic-elliptic Keller-Segel
equation. We finally employ a purely semigroup analysis to prove linear, and
then nonlinear, exponential stability of the system in appropriated functional
spaces with polynomial weights.
