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

Langhao Zhou; Liangwei Wang; Chunhua Jin

doi:10.3934/dcdsb.2021122

Article Contents

2022, Volume 27, Issue 4: 2065-2075. 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
^* Corresponding author: Liangwei Wang

Received: September 30, 2020

Revised: January 31, 2021

Early access: April 2021

Published: April 2022

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K55, 35B65.

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