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

Abstract

This paper investigates the following chemotaxis system featuring weak degradation and nonlinear motility functions \begin{equation}\label{Model1} \begin{cases} u_{t} = (γ(v)u)_{xx} + r - μu, & 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 ${\hat{I}}^3(v)$ satisfies the regularity conditions ${\hat{I}}^3\in C^2[0,\infty )$ with ${\hat{I}}^3(v)>0$ for all $v\ge 0$, and has bounded logarithmic derivative in the sense that $\sup_{v\geq 0} \frac{|γ'(v)|^{2}}{γ(v)} < \infty$. 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 $r,\hat{I}¼>0$ 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 $(r/\hat{I}¼,r/\hat{I}¼)$ in the parameter regime $\hat{I}¼>\frac{H_0}{16}$, where $H_{0} := \sup_{v \geq 0} \frac{|γ'(v)|^{2}}{γ(v)}$ 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.

19 pages, 9 figures