Global existence and boundedness of a chemotaxis model with indirect production and general kinetic function

Abstract

This paper is devoted to the chemotaxis model with indirect production and general kinetic function

$$\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+f(u),\qquad&x\in \Omega ,\,t>0,\\&v_t=\Delta v-v+w,\qquad&x\in \Omega ,\,t>0,\\&\tau w_t+\lambda w=g(u),\qquad&x\in \Omega ,\,t>0, \end{aligned} \right. \end{aligned}$$

in a bounded domain \(\Omega \subset \mathbb {R}^n(n\le 3)\) with smooth boundary \(\partial \Omega \), where \(\chi , \tau , \lambda \) are given positive parameters, f and g are known functions. We find several explicit conditions involving the kinetic function f, g, the parameters \(\chi \), \(\lambda \), and the initial data \(\Vert u_0\Vert _{L^1(\Omega )}\) to ensure the global-in-time existence and uniform boundedness for the corresponding 2D/3D Neumann initial-boundary value problem. Particularly, when \(f\equiv 0\), and g is a linear function, the global bounded classical solutions to the corresponding 2D Neumann initial-boundary value problem with arbitrarily large initial data and chemotactic sensitivity are established. Our results partially extend the results of Hu and Tao (Math Models Methods Appl Sci 26:2111–2128, 2016), Tao and Winkler (J Eur Math Soc 19:3641–3678, 2017), etc.

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Acknowledgements

The author would like to express her thanks to Professor Michael Winkler and the anonymous reviewer for their valuable suggestions, which greatly improved the exposition of this paper. This work was supported by the National Natural Science Foundation of China (No. 11701461) and the Postdoctoral Science Foundation of China (Nos. 2017M622990, 2018T110956).

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Correspondence to Xie Li.

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Li, X. Global existence and boundedness of a chemotaxis model with indirect production and general kinetic function. Z. Angew. Math. Phys. 71, 117 (2020). https://doi.org/10.1007/s00033-020-01339-z

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Mathematics Subject Classification

  • 35A01
  • 35B40
  • 35B45
  • 35Q92
  • 92C17

Keywords

  • Chemotaxis
  • Indirect signal production
  • Kinetic function
  • Global existence
  • Boundedness