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

Abstract

<p style='text-indent:20px;'>The paper is concerned with the following chemotaxis system with nonlinear motility functions <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\begin{equation}\label{0-1}\begin{cases}u_t = \nabla \cdot (\gamma(v)\nabla u- u\chi(v)\nabla v)+\mu u(1-u), &amp;x\in \Omega, ~~t&gt;0, \\ 0 = \Delta v+ u-v, &amp; x\in \Omega, ~~t&gt;0, \\u(x, 0) = u_0(x), &amp; x\in \Omega, \end{cases}~~~~(\ast)\end{equation}$ \end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>subject to homogeneous Neumann boundary conditions in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^2 $\end{document}</tex-math></inline-formula> with smooth boundary, where the motility functions <inline-formula><tex-math id="M2">\begin{document}$ \gamma(v) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \chi(v) $\end{document}</tex-math></inline-formula> satisfy the following conditions <p style='text-indent:20px;'>● <inline-formula><tex-math id="M4">\begin{document}$ (\gamma, \chi)\in [C^2[0, \infty)]^2 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M5">\begin{document}$ \gamma(v)&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ \frac{|\chi(v)|^2}{\gamma(v)} $\end{document}</tex-math></inline-formula> is bounded for all <inline-formula><tex-math id="M7">\begin{document}$ v\geq 0 $\end{document}</tex-math></inline-formula>. <p style='text-indent:20px;'>By employing the method of energy estimates, we establish the existence of globally bounded solutions of ($\ast$) with <inline-formula><tex-math id="M8">\begin{document}$ \mu&gt;0 $\end{document}</tex-math></inline-formula> for any <inline-formula><tex-math id="M9">\begin{document}$ u_0 \in W^{1, \infty}(\Omega) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M10">\begin{document}$ u_0 \geq (\not\equiv) 0 $\end{document}</tex-math></inline-formula>. Then based on a Lyapunov function, we show that all solutions <inline-formula><tex-math id="M11">\begin{document}$ (u, v) $\end{document}</tex-math></inline-formula> of ($\ast$) will exponentially converge to the unique constant steady state <inline-formula><tex-math id="M12">\begin{document}$ (1, 1) $\end{document}</tex-math></inline-formula> provided <inline-formula><tex-math id="M13">\begin{document}$ \mu&gt;\frac{K_0}{16} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M14">\begin{document}$ K_0 = \max\limits_{0\leq v \leq \infty}\frac{|\chi(v)|^2}{\gamma(v)} $\end{document}</tex-math></inline-formula>.