Unveiling Explicit Patterns: Exact Steady States and Stability in a Confined Chemotaxis Model

TL;DR

This paper introduces explicit steady-state solutions for a confined chemotaxis model, analyzing their stability and providing benchmarks for numerical studies, thereby advancing understanding of boundary-driven pattern formation.

Contribution

It presents a new family of explicit solutions for the Keller-Segel system on bounded intervals, including singular types, with rigorous stability analysis under biologically relevant conditions.

Findings

Explicit steady-state solutions derived using trigonometric and hyperbolic functions

Stability thresholds established via energy methods

Solutions serve as benchmarks for numerical validation

Abstract

Inspired by Carrillo-Li-Wang's work [Proc. London Math. Soc., 2021] on stationary solutions to the singular Keller-Segel system, this paper presents a novel family of explicit steady-state solutions for the same model on a bounded interval, expressed in terms of trigonometric and hyperbolic functions. Under Dirichlet boundary conditions and within a biologically stable parameter regime, these solutions, including singular types such as secant and cosecant, are rigorously derived and analyzed. Their stability is established via energy methods, yielding precise thresholds for pattern persistence. These results provide valuable benchmarks for numerical validation and offer insights into boundary-driven pattern formation.