Global existence and boundedness in a chemotaxis model with singular sensitivity and nonlocal term

The chemotaxis system

$$\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta u-\chi \nabla \cdot \left( \frac{u}{v}\nabla v\right) + u^{\alpha }\left( \gamma -\mu \int \limits _{\Omega }u^{\beta }\right) ,{} & {} x\in \Omega ,t>0,\\&v_t=\epsilon \Delta v-v+u,{} & {} x\in \Omega ,t>0, \end{aligned}\right. \end{aligned}$$

is considered under homogeneous Neumann boundary conditions in smoothly bounded domain \(\Omega \subseteq \mathbb {R}^n\), \(n\ge 2\), with constants \(0<\epsilon <1\), \(0<\chi <1-\epsilon \). It is asserted that the problem possesses a uniquely global classical solution whenever the numbers \(\alpha , \beta \) satisfy \(1<\alpha <2\), \(\beta >\frac{n}{2}+\alpha -1\) or \(\alpha \ge 2\), \(\beta >\frac{n}{2}(\alpha -1)+1\). Moreover, it is shown that if \(1<\alpha <2\), \(\beta >\max \{\frac{n}{2}+\alpha -1, \frac{(\alpha -1)(1-\epsilon )}{(2-\alpha )\chi }+1\}\) and \(\gamma >0\) is sufficiently large, then the global-in-time solution is uniformly bounded. In addition, we get similar results for the case of \(n=1\), which is worth mentioning that the requirement for \(\epsilon \) and \(\chi \) is very weak in the global existence result.