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Uniqueness of stationary states for singular Keller-Segel type models

Calvez, Vincent and Carrillo, José Antonio and Hoffmann, Franca (2021) Uniqueness of stationary states for singular Keller-Segel type models. Nonlinear Analysis, 205 . Art. No. 112222. ISSN 0362-546X.

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We consider a generalised Keller–Segel model with non-linear porous medium type diffusion and non-local attractive power law interaction, focusing on potentials that are more singular than Newtonian interaction. We show uniqueness of stationary states (if they exist) in any dimension both in the diffusion-dominated regime and in the fair-competition regime when attraction and repulsion are in balance. As stationary states are radially symmetric decreasing, the question of uniqueness reduces to the radial setting. Our key result is a sharp generalised Hardy–Littlewood–Sobolev type functional inequality in the radial setting.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Calvez, Vincent0000-0002-3674-1965
Carrillo, José Antonio0000-0001-8819-4660
Hoffmann, Franca0000-0002-1182-5521
Additional Information:© 2020 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license ( Received 20 September 2020, Accepted 2 December 2020, Available online 13 December 2020. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements No 639638 and No. 883363). JAC was partially supported by the EPSRC, UK through grant number EP/P031587/1. FH was partially supported by Caltech’s von Karman postdoctoral instructorship, USA .
Funding AgencyGrant Number
European Research Council (ERC)639638
European Research Council (ERC)883363
Engineering and Physical Sciences Research Council (EPSRC)EP/P031587/1
Caltech Von Kármán instructorshipUNSPECIFIED
Subject Keywords:uniqueness, Hardy-Littlewood-Sobolev inequality, aggregation-diffusion equation, Keller-Segel model
Classification Code:AMS Subject Classification: 35B38, 35B40, 26D10
Record Number:CaltechAUTHORS:20200331-073400209
Persistent URL:
Official Citation:Vincent Calvez, José Antonio Carrillo, Franca Hoffmann, Uniqueness of stationary states for singular Keller–Segel type models, Nonlinear Analysis, Volume 205, 2021, 112222, ISSN 0362-546X, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:102183
Deposited By: Tony Diaz
Deposited On:31 Mar 2020 16:41
Last Modified:31 Mar 2021 23:03

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