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

The paper is concerned with the following chemotaxis system with nonlinear motility functions

\begin{document}$\begin{equation}\label{0-1}\begin{cases}u_t = \nabla \cdot (\gamma(v)\nabla u- u\chi(v)\nabla v)+\mu u(1-u), &x\in \Omega, ~~t>0, \\ 0 = \Delta v+ u-v, & x\in \Omega, ~~t>0, \\u(x, 0) = u_0(x), & x\in \Omega, \end{cases}~~~~(\ast)\end{equation}$ \end{document}

subject to homogeneous Neumann boundary conditions in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^2 $\end{document} with smooth boundary, where the motility functions \begin{document}$ \gamma(v) $\end{document} and \begin{document}$ \chi(v) $\end{document} satisfy the following conditions

● \begin{document}$ (\gamma, \chi)\in [C^2[0, \infty)]^2 $\end{document} with \begin{document}$ \gamma(v)>0 $\end{document} and \begin{document}$ \frac{|\chi(v)|^2}{\gamma(v)} $\end{document} is bounded for all \begin{document}$ v\geq 0 $\end{document}.

By employing the method of energy estimates, we establish the existence of globally bounded solutions of ($\ast$) with \begin{document}$ \mu>0 $\end{document} for any \begin{document}$ u_0 \in W^{1, \infty}(\Omega) $\end{document} with \begin{document}$ u_0 \geq (\not\equiv) 0 $\end{document}. Then based on a Lyapunov function, we show that all solutions \begin{document}$ (u, v) $\end{document} of ($\ast$) will exponentially converge to the unique constant steady state \begin{document}$ (1, 1) $\end{document} provided \begin{document}$ \mu>\frac{K_0}{16} $\end{document} with \begin{document}$ K_0 = \max\limits_{0\leq v \leq \infty}\frac{|\chi(v)|^2}{\gamma(v)} $\end{document}.