On the self-similar stability of the parabolic-parabolic Keller–Segel equation

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.