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

In this paper, we shall study the initial-boundary value problem of a chemotaxis model with signal-dependent diffusion and sensitivity as follows\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} \;\;(*)$\end{document}in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^2 $\end{document} with smooth boundary, where \begin{document}$ \alpha,\beta, D $\end{document} are positive constants, \begin{document}$ \theta\in \mathbb{R} $\end{document} and \begin{document}$ \nu $\end{document} denotes the outward normal vector of \begin{document}$ \partial \Omega $\end{document} . The functions \begin{document}$ \chi(v),\gamma(v) $\end{document} and \begin{document}$ F(v) $\end{document} satisfy ● \begin{document}$ (\gamma(v),\chi(v))\in [C^2[0,\infty)]^2 $\end{document} with \begin{document}$ \gamma(v)>0,\gamma'(v)<0 $\end{document} and \begin{document}$ \frac{|\chi(v)|+|\gamma'(v)|}{\gamma(v)} $\end{document} is bounded; ● \begin{document}$ 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). $\end{document} We first prove that the existence of globally bounded solution of system (*) based on the method of weighted energy estimates. Moreover, by constructing Lyapunov functional, we show that the solution \begin{document}$ (u,v,w) $\end{document} will converge to \begin{document}$ (0,0,w_*) $\end{document} in \begin{document}$ L^\infty $\end{document} with some \begin{document}$ w_*\geq0 $\end{document} as time tends to infinity in the case of \begin{document}$ \theta\leq 0 $\end{document} , while if \begin{document}$ \theta>0 $\end{document} , the solution \begin{document}$ (u,v,w) $\end{document} will asymptotically converge to \begin{document}$ (\frac{\theta}{\beta},\frac{\theta}{\beta},0) $\end{document} in \begin{document}$ L^\infty $\end{document} -norm provided \begin{document}$ D>\max\limits_{0\leq v\leq \infty}\frac{\theta|\chi(v)|^2}{16\beta^2\gamma(v)} $\end{document} .