  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 $\Omega\subset\mathbb{R}^n$ $(n\ge2)$, where parameters
$r,\mu>0$ and density governed sensitivity fulfills $ S(u) \simeq
u(u+1)^{\beta-1}$ for all $u\ge0$ with $\beta\in \mathbb{R}$. It is proved that
for any $r,\mu>0$, there exists a global classical solution if $\beta<1$ and
$n\ge2$. 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.

Analysis of PDEs