Abstract
ABSTRACTThe 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<8π, 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 L4∕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.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
