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Global well-posedness in a chemotaxis system with oxygen consumption

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

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  • 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)}} $.

    Mathematics Subject Classification: Primary: 35K55, 35Q92, 35Q35; Secondary: 92C17.


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