Unveiling Explicit Patterns: Exact Steady States and Stability in a Confined Chemotaxis Model
Yue Huang, Ling Xue, Kun Zhao, Xiaoming Zheng
TL;DR
This study derives three explicit steady-state solutions for a Keller–Segel–type chemotaxis model on a bounded interval using a Cole–Hopf transform, yielding Power, Trigonometric, and Hyperbolic families with corresponding $\bar{u},\bar{v}$ profiles. It proves nonlinear asymptotic stability under a parameter regime $\mathsf{d}>0$, $\varepsilon>0$, $\chi>0$, $\mu>0$ with small $L^2$ perturbations (and allows partially large derivatives) via energy methods, establishing exponential decay toward the steady states. The authors identify sharp boundary-data constraints ensuring existence and stability, and they provide numerical benchmarks showing convergence to the analytic baselines when these constraints are met, while boundary violations yield qualitatively different patterns. The results offer a rigorous, benchmarkable view of boundary-driven pattern formation in confined chemotaxis and point to future work on multi-domain settings and optimal control in regenerative medicine and oncology.
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.