A quasilinear parabolic-parabolic chemotaxis model with logistic source and singular sensitivity

This paper deals with the dynamical properties of the quasilinear parabolic-parabolic chemotaxis system\begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_{t} = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(\frac{u}{v} \nabla v)+\mu u- \mu u^{2}, \, \, \, &x\in\Omega, \, \, \, t>0, \\ v_{t} = \Delta v-v+u, &x\in\Omega, \, \, \, t>0, \end{array} \right. \end{eqnarray*} $\end{document}under homogeneous Neumann boundary conditions in a convex bounded domain \begin{document}$ \Omega\subset\mathbb{R}^{n} $\end{document} , \begin{document}$ n\geq2 $\end{document} , with smooth boundary. \begin{document}$ \chi>0 $\end{document} and \begin{document}$ \mu>0 $\end{document} , \begin{document}$ D(u) $\end{document} is supposed to satisfy the behind properties\begin{document}$ \begin{equation*} \begin{split} D(u)\geq (u+1)^{\alpha} \, \, \, \text{with}\, \, \, \alpha>0. \end{split} \end{equation*} $\end{document}It is shown that there is a positive constant \begin{document}$ m_{*} $\end{document} such that\begin{document}$ \begin{equation*} \begin{split} \int_{\Omega}u\geq m_{*} \end{split} \end{equation*} $\end{document}for all \begin{document}$ t\geq0 $\end{document} . Moreover, we prove that the solution is globally bounded. Finally, it is asserted that the solution exponentially converges to the constant stationary solution \begin{document}$ (1, 1) $\end{document} .