In this paper we study the zero-flux chemotaxis-system \begin{ equation*} \begin{ cases} u_{ t}=\nabla \cdot ((u+1)^{ m-1} \nabla u-(u+1)^\alpha \chi(v)\nabla v) + ku-\mu u^2 & x\in \Omega, t>0, \\ v_{ t} = \Delta v-vu & x\in \Omega, t>0,\\ \end{ cases} \end{ equation*} $\Omega$ being a bounded and smooth domain of $\mathbb{ R}^n$, $n\geq 1$, and where $m,k \in \mathbb{ R}$, $\mu>0$ and $\alpha < \frac{ m+1}{ 2}$. For any $v\geq 0$ the chemotactic sensitivity function is assumed to behave as the prototype $\chi(v) = \frac{ \chi_0}{ (1+av)^2}$, with $a\geq 0$ and $\chi_0>0$. We prove that for nonnegative and sufficiently regular initial data $u(x,0)$ and $v(x,0),$ the corresponding initial-boundary value problem admits a global bounded classical solution provided $\mu$ is large enough.
Categories: math.DS math.AP