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

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 $.

Keywords

• boundedness
• p-laplacian diffusion
• logistic source
• chemotaxis-stokes