Global solvability to a singular chemotaxis-consumption model with fast and slow diffusion and logistic source

<p style='text-indent:20px;'>In this paper, we consider the following chemotaxis-consumption model with porous medium diffusion and singular sensitivity</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \left\{ \begin{aligned} &amp;u_{t} = \Delta u^{m}-\chi \mathrm{div}(\frac{u}{v}\nabla v)+\mu u(1-u), \\ &amp;v_{t} = \Delta v-u^{r}v, \end{aligned}\right. \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb R^N $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ N\ge 2 $\end{document}</tex-math></inline-formula>) with zero-flux boundary conditions. It is shown that if <inline-formula><tex-math id="M3">\begin{document}$ r&lt;\frac{4}{N+2} $\end{document}</tex-math></inline-formula>, for arbitrary case of fast diffusion (<inline-formula><tex-math id="M4">\begin{document}$ m\le 1 $\end{document}</tex-math></inline-formula>) and slow diffusion <inline-formula><tex-math id="M5">\begin{document}$ (m&gt;1) $\end{document}</tex-math></inline-formula>, this problem admits a locally bounded global weak solution. It is worth mentioning that there are no smallness restrictions on the initial datum and chemotactic coefficient.</p>