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

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.