# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021122
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## Global solvability to a singular chemotaxis-consumption model with fast and slow diffusion and logistic source

 1 College of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, 404100, China 2 School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

* Corresponding author: Liangwei Wang

Received  October 2020 Revised  February 2021 Early access April 2021

In this paper, we consider the following chemotaxis-consumption model with porous medium diffusion and singular sensitivity
 \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*}
in a bounded domain
 $\Omega\subset\mathbb R^N$
(
 $N\ge 2$
) with zero-flux boundary conditions. It is shown that if
 $r<\frac{4}{N+2}$
, for arbitrary case of fast diffusion (
 $m\le 1$
) and slow diffusion
 $(m>1)$
, 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.
Citation: Langhao Zhou, Liangwei Wang, Chunhua Jin. Global solvability to a singular chemotaxis-consumption model with fast and slow diffusion and logistic source. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021122
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