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

Johannes Lankeit; Yulan Wang

doi:10.3934/dcds.2017262

Article Contents

2017, Volume 37, Issue 12: 6099-6121. Doi: 10.3934/dcds.2017262

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

Johannes Lankeit^1, , and
Yulan Wang^2,

1.
Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany
2.
School of Science, Xihua University, Chengdu 610039, China

^* Corresponding author: Johannes Lankeit
^* Corresponding author: Johannes Lankeit

Received: August 2016

Revised: July 2017

Published: October 2017

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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Abstract

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.

Keywords:

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

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