On an exponentially decaying diffusive chemotaxis system with indirect signals

This paper deals with an exponentially decaying diffusive chemotaxis system with indirect signal production or consumption

\begin{document}$ \begin{eqnarray*} \label{1a} \left\{ \begin{split}{} &u_t = \nabla\cdot(D(u)\nabla u)-\nabla\cdot(S(u)\nabla v), &(x,t)\in \Omega\times (0,\infty), \\ &v_t = \Delta v+h(v,w), &(x,t)\in \Omega\times (0,\infty), \\ &w_t = \Delta w- w+u, &(x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*} $\end{document}

under homogeneous Neumann boundary conditions in a smoothly bounded domain \begin{document}$ \Omega\subset \mathbb{R}^{n} $\end{document}, \begin{document}$ n\geq2 $\end{document}, where the nonlinear diffusivity \begin{document}$ D $\end{document} and chemosensitivity \begin{document}$ S $\end{document} are supposed to satisfy

\begin{document}$ K_{1}e^{-\beta^{-}s}\leq D(s) \leq K_{2}e^{-\beta^{+}s} \;\;\;{\rm{and}}\;\;\;\frac{D(s)}{S(s)}\geq K_{3}s^{-\alpha}+\gamma, $\end{document}

with the constants \begin{document}$ \beta^{-}\geq \beta^{+}>0 $\end{document}, \begin{document}$ K_{1},K_{2},K_{3}>0 $\end{document} and \begin{document}$ \alpha,\gamma\geq0 $\end{document}. When \begin{document}$ h(v,w) = -v+w $\end{document}, we study the global existence and boundedness of solutions for the above system provided that \begin{document}$ \alpha\in[0,\frac{2}{n}) $\end{document}, \begin{document}$ \beta^{-}\geq \beta^{+}>\frac{n}{2} $\end{document}, \begin{document}$ \gamma>1 $\end{document} and the initial mass of \begin{document}$ u_{0} $\end{document} is small enough. Moreover, it is proved that the global bounded solution \begin{document}$ (u,v,w) $\end{document} converges to \begin{document}$ (\overline{u_{0}},\overline{u_{0}},\overline{u_{0}}) $\end{document} in the \begin{document}$ L^{\infty} $\end{document}-norm as \begin{document}$ t\rightarrow \infty $\end{document}, where \begin{document}$ \overline{u_{0}} = \frac{1}{|\Omega|}\int_{\Omega}u_{0}(x)dx $\end{document}. When \begin{document}$ h(v,w) = -vw $\end{document}, it is shown that this system possesses a unique uniformly bounded classical solution if \begin{document}$ \alpha\geq0 $\end{document}, \begin{document}$ \gamma>0 $\end{document} and \begin{document}$ \beta^{-}\geq \beta^{+}>\frac{n}{2} $\end{document}. Furthermore, if \begin{document}$ n = 2 $\end{document}, \begin{document}$ \alpha\geq0 $\end{document}, \begin{document}$ \gamma\geq0 $\end{document}, and \begin{document}$ \beta^{-}\geq \beta^{+}>\varepsilon $\end{document} with some \begin{document}$ \varepsilon>0 $\end{document}, we only obtain the global existence of solutions for the above system.