The Keller-Segel system with logistic growth and signal-dependent motility

The paper is concerned with the following chemotaxis system nonlinear motility functions style='text-indent:20px;'> \begin{document}$\begin{equation}\label{0-1}\begin{cases}u_t = \nabla \cdot (\gamma(v)\nabla u- u\chi(v)\nabla v)+\mu u(1-u), &x\in \Omega, ~~t>0, \\ 0 \Delta v+ u-v, & x\in \\u(x, 0) u_0(x), \end{cases}~~~~(\ast)\end{equation}$ \end{document} style='text-indent:20px;'>subject to homogeneous Neumann boundary conditions in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^2 $\end{document} smooth boundary, where id="M2">\begin{document}$ \gamma(v) and id="M3">\begin{document}$ \chi(v) satisfy style='text-indent:20px;'>? id="M4">\begin{document}$ (\gamma, \chi)\in [C^2[0, \infty)]^2 id="M5">\begin{document}$ \gamma(v)>0 id="M6">\begin{document}$ \frac{|\chi(v)|^2}{\gamma(v)} for all id="M7">\begin{document}$ v\geq $\end{document}. style='text-indent:20px;'>By employing method of energy estimates, we establish existence globally solutions ($\ast$) id="M8">\begin{document}$ \mu>0 any id="M9">\begin{document}$ u_0 \in W^{1, \infty}(\Omega) id="M10">\begin{document}$ \geq (\not\equiv) Then based on Lyapunov function, show that id="M11">\begin{document}$ (u, v) will exponentially converge unique constant steady state id="M12">\begin{document}$ (1, 1) provided id="M13">\begin{document}$ \mu>\frac{K_0}{16} id="M14">\begin{document}$ K_0 \max\limits_{0\leq v \leq \infty}\frac{|\chi(v)|^2}{\gamma(v)}