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

作者：Carrapatoso K; Mischler S

来源：Communications in Partial Differential Equations, 2017, 42(2): 291-345.

DOI：10.1080/03605302.2017.1280682

• Keller-Segel system
• long-time behavior
• regularization
• self-similar variables
• stability
• uniqueness

The present paper deals with the parabolic-parabolic Keller-Segel equation in the plane in the general framework of weak (or free energy) solutions associated to initial data with finite mass M<8, 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 exponential stability of the self-similar profile in the quasiparabolic-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.

• 出版日期2017