Global bounded solution of a 3D chemotaxis-Stokes system with slow $ p $-Laplacian diffusion and logistic source

Xindan Zhou; Zhongping Li; Xindan Zhou; Zhongping Li

doi:10.3934/math.2024782

AIMS Mathematics

2024, Volume 9, Issue 6: 16168-16186. doi: 10.3934/math.2024782

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Research article

Global bounded solution of a 3D chemotaxis-Stokes system with slow $ p $-Laplacian diffusion and logistic source

Xindan Zhou ^1,2,
Zhongping Li ^{1,2
,
,}

1.
College of Mathematic and Information, China West Normal University, Nanchong, 637009, China
2.
Sichuan Colleges and Universities Key Laboratory of Optimization Theory and Applications, School of Mathematics and Information, China West Normal University, Nanchong, 637009, China

Received: 27 January 2024 Revised: 02 April 2024 Accepted: 09 April 2024 Published: 08 May 2024
MSC : 35K55, 35B35

In this paper, the chemotaxis-Stokes system with slow $ p $-Laplacian diffusion and logistic source as follows

$ \begin{equation*} \left\{ \begin{aligned} &n_t+u\cdot\nabla n = \nabla\cdot(|\nabla n|^{p-2}\nabla n)-\nabla\cdot(n\nabla c)+\mu n(1-n), &x\in\Omega, t>0, \\ &c_t+u\cdot\nabla c = \Delta c-cn, & x\in\Omega, t>0, \\ &u_t+\nabla P = \Delta u+n\nabla\Phi, & x\in\Omega, t>0, \\ &\nabla\cdot u = 0, &\; x\in\Omega, t>0\; \end{aligned} \right. \end{equation*} $

was considered in a bounded domain $ \Omega\subset\mathbb{R}^3 $ with smooth boundary under homogeneous Neumann-Neumann-Dirichlet boundary conditions. Subject to the effect of logistic source, we proved the system exists a global bounded weak solution for any $ p > 2 $.
- boundedness,
- $ p $-Laplacian diffusion,
- logistic source,
- chemotaxis-Stokes
Citation: Xindan Zhou, Zhongping Li. Global bounded solution of a 3D chemotaxis-Stokes system with slow $ p $-Laplacian diffusion and logistic source[J]. AIMS Mathematics, 2024, 9(6): 16168-16186. doi: 10.3934/math.2024782

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Abstract

In this paper, the chemotaxis-Stokes system with slow $ p $-Laplacian diffusion and logistic source as follows

$ \begin{equation*} \left\{ \begin{aligned} &n_t+u\cdot\nabla n = \nabla\cdot(|\nabla n|^{p-2}\nabla n)-\nabla\cdot(n\nabla c)+\mu n(1-n), &x\in\Omega, t>0, \\ &c_t+u\cdot\nabla c = \Delta c-cn, & x\in\Omega, t>0, \\ &u_t+\nabla P = \Delta u+n\nabla\Phi, & x\in\Omega, t>0, \\ &\nabla\cdot u = 0, &\; x\in\Omega, t>0\; \end{aligned} \right. \end{equation*} $

was considered in a bounded domain $ \Omega\subset\mathbb{R}^3 $ with smooth boundary under homogeneous Neumann-Neumann-Dirichlet boundary conditions. Subject to the effect of logistic source, we proved the system exists a global bounded weak solution for any $ p > 2 $.

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