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{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*} $

under homogeneous Neumann boundary conditions in a convex bounded domain $ \Omega\subset\mathbb{R}^{n} $, $ n\geq2 $, with smooth boundary. $ \chi>0 $ and $ \mu>0 $, $ D(u) $ is supposed to satisfy the behind properties

$ \begin{equation*} \begin{split} D(u)\geq (u+1)^{\alpha} \, \, \, \text{with}\, \, \, \alpha>0. \end{split} \end{equation*} $

It is shown that there is a positive constant $ m_{*} $ such that

$ \begin{equation*} \begin{split} \int_{\Omega}u\geq m_{*} \end{split} \end{equation*} $

for all $ t\geq0 $. Moreover, we prove that the solution is globally bounded. Finally, it is asserted that the solution exponentially converges to the constant stationary solution $ (1, 1) $.