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

Abstract

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 behaviour of a population of bacteria that interact by means of a signalling substance. We assume an external confinement for the bacteria and a nonlinear dependency of the chemotactic drift on the signalling substance concentration. We perform an analysis of existence and long-time behaviour 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.

Improved results: nonlinearity also comprises classical Keller-Segel model; moreover: references added, typos corrected

References in corpus (2)

• Large mass self-similar solutions of the parabolic-parabolic Keller--Segel model of chemotaxis
• A gradient flow approach to a thin film approximation of the Muskat problem

Cited by in corpus (5)

• Incompressible immiscible multiphase flows in porous media: a variational approach
• Exponential convergence to equilibrium in a Poisson-Nernst-Planck-type system with nonlinear diffusion
• A note on the variational analysis of the parabolic-parabolic Keller-Segel system in one spatial dimension
• High-frequency limit of non-autonomous gradient flows
• A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations