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Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption

  • * Corresponding author: Johannes Lankeit

    * Corresponding author: Johannes Lankeit 
J. Lankeit acknowledges support of the Deutsche Forschungsgemeinschaft within the project Analysis of chemotactic cross-diffusion in complex frameworks. Y. Wang was supported by the NNSF of China (no. 11501457).
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  • This paper deals with the homogeneous Neumann boundary-value problem for the chemotaxis-consumption system

    $\left\{ \begin{align} & {{u}_{t}}=\Delta u-\chi \nabla \cdot \left( u\nabla v \right)+\kappa u-\mu {{u}^{2}},\ \ \ \ \ \ \ x\in \mathit{\Omega },t>0, \\ & {{v}_{t}}=\Delta v-uv,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in \mathit{\Omega },t>0, \\ \end{align} \right.$

    in $N$-dimensional bounded smooth domains for suitably regular positive initial data.

    We shall establish the existence of a global bounded classical solution for suitably large $μ$ and prove that for any $μ>0$ there exists a weak solution.

    Moreover, in the case of $κ>0$ convergence to the constant equilibrium $(\frac{κ}{μ },0)$ is shown.

    Mathematics Subject Classification: 35Q92, 35K55, 35A01, 35B40, 35D30, 92C17.


    \begin{equation} \\ \end{equation}
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