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Uniqueness and long time asymptotics for the parabolic-parabolic Keller-Segel equation. (English) Zbl 1377.35134
Summary: 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 \(M<8\pi\), 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 \(L^{4}\) Lebesgue norm similarly as for the \(2d\) vorticity Navier-Stokes equation. (2) We prove a radially symmetric and polynomial weighted \(H^1 \times H^2\) 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.

MSC:
35K45 Initial value problems for second-order parabolic systems
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
35B60 Continuation and prolongation of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
92B05 General biology and biomathematics
92C17 Cell movement (chemotaxis, etc.)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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