Uniqueness for Keller-Segel-type chemotaxis models

José Antonio Carrillo; Stefano Lisini; Edoardo Mainini

doi:10.3934/dcds.2014.34.1319

Article Contents

2014, Volume 34, Issue 4: 1319-1338. Doi: 10.3934/dcds.2014.34.1319

This issue Previous Article Optimal location problems with routing cost Next Article On the twist condition and $c$-monotone transport plans

Uniqueness for Keller-Segel-type chemotaxis models

1.
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ
2.
Università degli Studi di Pavia, Dipartimento di Matematica “F. Casorati”, via Ferrata 1, 27100 Pavia
3.
Dipartimento di Ingegneria meccanica, energetica, gestionale e dei trasporti (DIME), Università degli Studi di Genova, P.le Kennedy 1, 16129 Genova

Received: December 2012

Revised: April 2013

Published: April 2014

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Abstract

We prove uniqueness in the class of integrable and bounded nonnegative solutions in the energy sense to the Keller-Segel (KS) chemotaxis system. Our proof works for the fully parabolic KS model, it includes the classical parabolic-elliptic KS equation as a particular case, and it can be generalized to nonlinear diffusions in the particle density equation as long as the diffusion satisfies the classical McCann displacement convexity condition. The strategy uses Quasi-Lipschitz estimates for the chemoattractant equation and the above-the-tangent characterizations of displacement convexity. As a consequence, the displacement convexity of the free energy functional associated to the KS system is obtained from its evolution for bounded integrable initial data.

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Mathematics Subject Classification: 35A02, 35K45, 35Q92.

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