Vol. 8, No. 2, 2015

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Exponential convergence to equilibrium in a coupled gradient flow system modeling chemotaxis

Jonathan Zinsl and Daniel Matthes

Vol. 8 (2015), No. 2, 425–466

We study a system of two coupled nonlinear parabolic equations. It constitutes a variant of the Keller–Segel model for chemotaxis; i.e., it models the behavior of a population of bacteria that interact by means of a signaling substance. We assume an external confinement for the bacteria and a nonlinear dependency of the chemotactic drift on the signaling substance concentration.

We perform an analysis of existence and long-time behavior of solutions based on the underlying gradient flow structure of the system. The result is that, for a wide class of initial conditions, weak solutions exist globally in time and converge exponentially fast to the unique stationary state under suitable assumptions on the convexity of the confinement and the strength of the coupling.

gradient flow, Wasserstein metric, chemotaxis
Mathematical Subject Classification 2010
Primary: 35K45
Secondary: 35A15, 35B40, 35D30, 35Q92
Received: 12 May 2014
Revised: 25 November 2014
Accepted: 9 January 2015
Published: 10 May 2015
Jonathan Zinsl
Zentrum für Mathematik
Technische Universität München
Boltzmannstraße 3
85747 Garching
Daniel Matthes
Zentrum für Mathematik
Technische Universität München
Boltzmannstraße 3
85747 Garching