On a chemotaxis model with singular sensitivity: Convergence rate towards spiky steady state

Zefu Feng; Yao Tang; Wanwan Wang; Kun Zhao; Neng Zhu

doi:10.3934/dcds.2024165

Article Contents

2025, Volume 45, Issue 7: 2281-2301. Doi: 10.3934/dcds.2024165

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On a chemotaxis model with singular sensitivity: Convergence rate towards spiky steady state

1.
School of Mathematical Science, Chongqing Normal University, Chongqing 401331, China
2.
Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada
3.
School of Mathematics and Information Science, Nanchang Hangkong University, Nanchang 330063, China
4.
College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, China
5.
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA
6.
Department of Mathematics & Institute of Mathematics and Interdisciplinary Sciences, Nanchang University, Nanchang 330031, China

^*Corresponding author: Kun Zhao
^*Corresponding author: Kun Zhao

Received: June 2024

Revised: October 2024

Early access: December 9, 2024

Published online: December 9, 2024

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Abstract

This paper is concerned with the stability analysis for a model of chemotaxis:

$ \begin{equation*} \begin{aligned} u_t & = u_{xx} - \chi[u(\ln w)_x]_x, &(x, t)&\in \mathbb{R}_+ \times \mathbb{R}_+, \\ w_t & = \varepsilon w_{xx} - u^\gamma w^m, &(x, t)&\in \mathbb{R}_+ \times \mathbb{R}_+. \end{aligned} \end{equation*} $

By utilizing spatially weighted energy method, it is shown that the non-trivial steady state associated with the model under certain parametric constraints and the boundary conditions:

$ \begin{equation*} \begin{aligned} &u_x = \chi u(\ln w)_x, \ \ w = b>0, &&(x, t) \in \{0\}\times \mathbb{R}_+, \\ &(u, w) \to (0, 0), && x\to\infty, \ t\in \mathbb{R}_+, \end{aligned} \end{equation*} $

is locally asymptotically stable. By exploring the idea of "gaining temporal integrability through faster spatial decay", the explicit (algebraic) convergence rate, in terms of the system parameters, of general solutions of the initial-boundary value problem towards the steady state is identified.

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Mathematics Subject Classification: Primary: 35Q92, 35K51, 35B35, 35B40.

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References

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