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

The chemotaxis system \(\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta u-\chi \nabla \cdot \left( \frac{u}{v}\nabla v\right) + u^{\alpha }\left( \gamma -\mu \int \limits _{\Omega }u^{\beta }\right) ,{} & {} x\in \Omega ,t>0,\\&v_t=\epsilon \Delta v-v+u,{} & {} x\in \Omega ,t>0, \end{aligned}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}& u_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }·\left(\frac{u}{v},\mathrm{\nabla },v\right)+u^{\alpha }\left(\gamma ,-,\mu ,\underset{\mathrm{\Omega }}{\int },u^{\beta }\right),\hfill & \hfill x\in \mathrm{\Omega },t>0,\\ \hfill & v_t=ϵ\mathrm{\Delta }v-v+u,\hfill & \hfill x\in \mathrm{\Omega },t>0,\end{array}\right)\end{array}$$ is considered under homogeneous Neumann boundary conditions in smoothly bounded domain \(\Omega \subseteq \mathbb {R}^n\) $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ , \(n\ge 2\) $n\ge 2$ , with constants \(0<\epsilon <1\) $0<ϵ<1$ , \(0<\chi <1-\epsilon \) $0<\chi <1-ϵ$ . It is asserted that the problem possesses a uniquely global classical solution whenever the numbers \(\alpha , \beta \) $\alpha ,\beta$ satisfy \(1<\alpha <2\) $1<\alpha <2$ , \(\beta >\frac{n}{2}+\alpha -1\) $\beta >\frac{n}{2}+\alpha -1$ or \(\alpha \ge 2\) $\alpha \ge 2$ , \(\beta >\frac{n}{2}(\alpha -1)+1\) $\beta >\frac{n}{2}(\alpha -1)+1$ . Moreover, it is shown that if \(1<\alpha <2\) $1<\alpha <2$ , \(\beta >\max \{\frac{n}{2}+\alpha -1, \frac{(\alpha -1)(1-\epsilon )}{(2-\alpha )\chi }+1\}\) $\beta >max\{\frac{n}{2}+\alpha -1,\frac{(\alpha -1)(1-ϵ)}{(2-\alpha )\chi }+1\}$ and \(\gamma >0\) $\gamma >0$ is sufficiently large, then the global-in-time solution is uniformly bounded. In addition, we get similar results for the case of \(n=1\) $n=1$ , which is worth mentioning that the requirement for \(\epsilon \) $ϵ$ and \(\chi \) $\chi$ is very weak in the global existence result.