In this paper, we consider the following chemotaxis-consumption model with porous medium diffusion and singular sensitivity \begin{document}$ \begin{align*} \left\{ \begin{aligned} &u_{t} = \Delta u^{m}-\chi \mathrm{div}(\frac{u}{v}\nabla v)+\mu u(1-u), \\ &v_{t} = \Delta v-u^{r}v, \end{aligned}\right. \end{align*} $\end{document} in a bounded domain \begin{document}$ \Omega\subset\mathbb R^N $\end{document} (\begin{document}$ N\ge 2 $\end{document}) with zero-flux boundary conditions. It is shown that if \begin{document}$ r<\frac{4}{N+2} $\end{document}, for arbitrary case of fast diffusion (\begin{document}$ m\le 1 $\end{document}) and slow diffusion \begin{document}$ (m>1) $\end{document}, 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.

We discuss the $\omega$-limit set for the Cauchy problem of the porous medium equation with initial data in some weighted spaces. Exactly, we show that there exists some relationship between the $\omega$-limit set of the rescaled initial data and the $\omega$-limit set of the spatially rescaled version of solutions. We also give some applications of such a relationship.