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February  2022, 21(2): 471-492. doi: 10.3934/cpaa.2021184

## Global well-posedness in a chemotaxis system with oxygen consumption

 School of Mathematical Sciences, University of Electronic, Science and Technology of China, Chengdu, 611731, China

Received  May 2021 Revised  August 2021 Published  February 2022 Early access  November 2021

Fund Project: This work is supported by the Applied Fundamental Research Program of Sichuan Province (no. 2020YJ0264)

Motivated by the studies of the hydrodynamics of the tethered bacteria Thiovulum majus in a liquid environment, we consider the following chemotaxis system
 $\begin{equation*} \left\{ \begin{split} & n_t = \Delta n-\nabla\cdot\left(n\chi(c)\nabla{c}\right)+nc, &x\in \Omega, t>0, \ & c_t = \Delta c-{\bf u}\cdot\nabla c-nc, &x\in \Omega, t>0\ \end{split} \right. \end{equation*}$
under homogeneous Neumann boundary conditions in a bounded convex domain
 $\Omega\subset \mathbb{R}^d(d\in\{2, 3\})$
with smooth boundary. For any given fluid
 ${\bf u}$
, it is proved that if
 $d = 2$
, the corresponding initial-boundary value problem admits a unique global classical solution which is uniformly bounded, while if
 $d = 3$
, such solution still exists under the additional condition that
 $0<\chi\leq \frac{1}{16\|c(\cdot, 0)\|_{L^\infty(\Omega)}}$
.
Citation: Xujie Yang. Global well-posedness in a chemotaxis system with oxygen consumption. Communications on Pure & Applied Analysis, 2022, 21 (2) : 471-492. doi: 10.3934/cpaa.2021184
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