Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity

Abstract

<p style='text-indent:20px;'>In this paper, we shall study the initial-boundary value problem of a chemotaxis model with signal-dependent diffusion and sensitivity as follows</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE8011"> \begin{document}$ \begin{cases} u_t = \nabla\cdot(\gamma(v)\nabla u-\chi(v)u\nabla v)+\alpha u F(w) +\theta u-\beta u^2, &x\in \Omega, \; \; t>0,\\ v_t = D\Delta v+u-v,& x\in \Omega, \; \; t>0,\\ w_t = \Delta w-uF(w),& x\in \Omega, \; \; t>0,\\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0,&x\in \partial\Omega, \; \; t>0,\\ u(x,0) = u_0(x), v(x,0) = v_0(x),w(x,0) = w_0(x), & x\in\Omega, \end{cases} \;\;(*)$ \end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^2 $\end{document}</tex-math></inline-formula> with smooth boundary, where <inline-formula><tex-math id="M2">\begin{document}$ \alpha,\beta, D $\end{document}</tex-math></inline-formula> are positive constants, <inline-formula><tex-math id="M3">\begin{document}$ \theta\in \mathbb{R} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \nu $\end{document}</tex-math></inline-formula> denotes the outward normal vector of <inline-formula><tex-math id="M5">\begin{document}$ \partial \Omega $\end{document}</tex-math></inline-formula>. The functions <inline-formula><tex-math id="M6">\begin{document}$ \chi(v),\gamma(v) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ F(v) $\end{document}</tex-math></inline-formula> satisfy</p><p style='text-indent:20px;'>● <inline-formula><tex-math id="M8">\begin{document}$ (\gamma(v),\chi(v))\in [C^2[0,\infty)]^2 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M9">\begin{document}$ \gamma(v)>0,\gamma'(v)<0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ \frac{|\chi(v)|+|\gamma'(v)|}{\gamma(v)} $\end{document}</tex-math></inline-formula> is bounded;</p><p style='text-indent:20px;'>● <inline-formula><tex-math id="M11">\begin{document}$ F(w)\in C^1([0,\infty)), F(0) = 0,F(w)>0 \ \mathrm{in}\; (0,\infty)\; \mathrm{and}\; F'(w)>0 \ \mathrm{on}\ \ [0,\infty). $\end{document}</tex-math></inline-formula></p><p style='text-indent:20px;'>We first prove that the existence of globally bounded solution of system (*) based on the method of weighted energy estimates. Moreover, by constructing Lyapunov functional, we show that the solution <inline-formula><tex-math id="M12">\begin{document}$ (u,v,w) $\end{document}</tex-math></inline-formula> will converge to <inline-formula><tex-math id="M13">\begin{document}$ (0,0,w_*) $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M14">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula> with some <inline-formula><tex-math id="M15">\begin{document}$ w_*\geq0 $\end{document}</tex-math></inline-formula> as time tends to infinity in the case of <inline-formula><tex-math id="M16">\begin{document}$ \theta\leq 0 $\end{document}</tex-math></inline-formula>, while if <inline-formula><tex-math id="M17">\begin{document}$ \theta>0 $\end{document}</tex-math></inline-formula>, the solution <inline-formula><tex-math id="M18">\begin{document}$ (u,v,w) $\end{document}</tex-math></inline-formula> will asymptotically converge to <inline-formula><tex-math id="M19">\begin{document}$ (\frac{\theta}{\beta},\frac{\theta}{\beta},0) $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M20">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula>-norm provided <inline-formula><tex-math id="M21">\begin{document}$ D>\max\limits_{0\leq v\leq \infty}\frac{\theta|\chi(v)|^2}{16\beta^2\gamma(v)} $\end{document}</tex-math></inline-formula>.</p>