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

Kleber Carrapatoso (1) , Stéphane Mischler (2)

1 CMLA - Centre de Mathématiques et de Leurs Applications
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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 $M< 8\pi$, 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.

Mots clés: Keller-Segel system; uniqueness; regularization; self-similar variables; long-time behaviour; stability

Domaines: Equations aux dérivées partielles [math.AP]

Kleber Carrapatoso, Stéphane Mischler. Uniqueness and long time asymptotics for the parabolic-parabolic Keller-Segel equation. Communications in Partial Differential Equations, 2017, 42 (2), pp.291-345. ⟨10.1080/03605302.2017.1280682⟩. ⟨hal-01011361v4⟩