Global existence, boundedness and asymptotic behavior to a logistic chemotaxis model with density-signal governed sensitivity and signal absorption

In present paper, we consider a chemotaxis consumption system with density-signal governed sensitivity and logistic source: $u_t=\Delta u-\nabla\cdot(\frac{S(u)}{v}\nabla v)+ru-\mu u^2$,$v_t=\Delta v-uv$ in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ $(n\ge 2)$, where parameters $r,\mu >0$ and density governed sensitivity fulfills $ S(u) \simeq u(u+1)^{\beta-1}$forall$u\ge0$with$\beta\in $\mathbb{R}$$. It is proved that for any $r,\mu >0$, there exists a global classical solution if $\beta <1$ and $n\ge 2$. Moreover, the global boundedness and the asymptotic behavior of the classical solution are determined for the case $\beta \in [0,1)$ in two dimensional setting, that is, the global solution $(u,v)$ is uniformly bounded in time and $\Big(u,v,\frac{|\nabla v|}{v}\Big)\longrightarrow\Big(\frac{r}{\mu},0,0\Big)~in~L^{\infty}(\Omega)~as~t\rightarrow\infty$, provided $\mu$ sufficiently large.