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

Abstract

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 $\Omega \subset {\mathbb{R}}^n$ $(n\ge 2)$, where parameters $r,\mu >0$ and density governed sensitivity fulfills $S(u)\simeq u(u+1{)}^{\beta -1}$ for all $u\ge 0$ 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 $(u,v,\frac{|\nabla v|}{v})⟶(\frac{r}{\mu },0,0)in{L}^{\infty }(\Omega )ast\to \infty$, provided $\mu$ sufficiently large.