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

Abstract : 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.
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Cited literature [54 references]

https://hal.archives-ouvertes.fr/hal-01011361
Contributor : Kleber Carrapatoso <>
Submitted on : Wednesday, December 21, 2016 - 5:15:00 PM
Last modification on : Thursday, April 15, 2021 - 3:31:51 AM
Long-term archiving on: : Tuesday, March 21, 2017 - 12:08:14 AM

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### Identifiers

• HAL Id : hal-01011361, version 4
• ARXIV : 1406.6006

### Citation

Kleber Carrapatoso, Stéphane Mischler. Uniqueness and long time asymptotics for the parabolic-parabolic Keller-Segel equation. 2016. ⟨hal-01011361v4⟩

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