Asymptotic stability of exogenous chemotaxis systems with physical boundary conditions

TL;DR

This paper proves the global existence and exponential stability of solutions to a one-dimensional exogenous chemotaxis system with mixed boundary conditions, overcoming previous analytical challenges.

Contribution

It introduces a novel anti-derivative technique to handle Dirichlet boundary conditions, establishing the first global well-posedness and asymptotic stability results for this system.

Findings

Global strong solutions exist and are unique.

Solutions exponentially stabilize to stationary states.

First such results for chemotaxis systems with physical boundary conditions.

Abstract

In this paper, we consider the exogenous chemotaxis system with physical mixed zero-flux and Dirichlet boundary conditions in one dimension. Since the Dirichlet boundary condition can not contribute necessary estimates for the cross-diffusion structure in the system, the global-in-time existence and asymptotic behavior of solutions remain open up to date. In this paper, we overcome this difficulty by employing the technique of taking anti-derivative so that the Dirichlet boundary condition can be fully used, and show that the system admits global strong solutions which exponentially stabilize to the unique stationary solution as time tends to infinity against some suitable small perturbations. To the best of our knowledge, this is the first result obtained on the global well-posedness and asymptotic behavior of solutions to the exogenous chemotaxis system with physical boundary conditions.