Vol. 8, No. 2, 2015

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 5, 1501–1870
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
Editors' interests
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author index
To appear
Other MSP journals
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