Global solutions in a chemotaxis consumption model with singular sensitivity

Abstract

In this paper, we consider the global existence of solutions to a chemotaxis consumption model with singular sensitivity

$$\begin{aligned} \begin{aligned} \left\{ {\begin{array}{*{20}{l}} {{u_t} = \nabla \cdot (D(u)\nabla u) - \nabla \cdot \left( {\frac{u}{v}\nabla v} \right) ,}&{}\quad {x \in \Omega ,t> 0,} \\ {{v_t} = \Delta v - u^{\beta }v,}&{}\quad {x \in \Omega ,t > 0,} \end{array}} \right. \end{aligned} \end{aligned}$$

in a smooth bounded domain $\Omega \subset {\mathbb {R}^{n}}$ $(n\ge 2)$ with the homogeneous Neumann boundary conditions, where the parameter $\beta >0$ and $D(u)\ge au^{m-1}$ with $m\ge 1$, $a>0$. It is proved that for any sufficiently regular initial data, this system possesses a global classical solution provided that $\beta <\frac{2}{n}$ or $m>1+\frac{n\beta }{4}$.