Global attractor of chemotaxis system with weak degradation and density-dependent motion

TL;DR

This paper analyzes a chemotaxis system with weak degradation and nonlinear motility, proving global boundedness, exponential convergence to equilibrium, and supporting findings with numerical simulations.

Contribution

It establishes the global existence, boundedness, and exponential convergence of solutions for a chemotaxis model with weak degradation and nonlinear motility functions, including numerical validation.

Findings

Solutions are globally bounded for all positive parameters.

Solutions converge exponentially to a unique equilibrium under certain conditions.

Numerical simulations confirm theoretical results and explore long-term behaviors.

Abstract

This paper investigates the following chemotaxis system featuring weak degradation and nonlinear motility functions \begin{equation}\label{Model1} \begin{cases} u_{t} = (\gamma(v)u)_{xx} + r - \mu u, & x \in [0,L],\ t > 0, v_{t} = v_{xx} - v + u, & x \in [0,L],\ t > 0, \end{cases} \end{equation} defined on the bounded interval $[0,L]$ with homogeneous Neumann boundary conditions. The motility function $\gamma(v)$ satisfies the regularity conditions $\gamma \in C^{2}[0,\infty)$ with $\gamma(v) > 0$ for all $v \geq 0$, and has bounded logarithmic derivative in the sense that $\sup_{v\geq 0} \frac{|\gamma'(v)|^{2}}{\gamma(v)} < \infty$. Our main results establish three fundamental properties of the system. Firstly, using energy estimate methods, we prove the existence of globally bounded solutions for all positive parameters $r, \mu > 0$ and non-negative, non-trivial initial data $u_{0} \in W^{1,\infty}([0,L])$. Secondly, through the construction of an appropriate Lyapunov function, we demonstrate that all solutions $(u,v)$ converge exponentially to the unique constant equilibrium $(r/\mu, r/\mu)$ in the parameter regime $\mu > \frac{H_{0}}{16}$, where $H_{0} := \sup_{v \geq 0} \frac{|\gamma'(v)|^{2}}{\gamma(v)}$ quantifies the maximal relative variation of the motility function. Finally, we present numerical results that not only validate the theoretical findings but also investigate the long-term behavior of solutions under diverse parameter configurations and initial conditions in two- and three-dimensional domains, providing valuable benchmarks for future research.