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On a generalized volume-filling chemotaxis system with nonlinear signal production

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Abstract

This paper deals with a generalized volume-filling chemotaxis system with nonlinear signal production

$$\begin{aligned} \left\{ \begin{aligned}&u_t=\nabla \cdot (\varphi (u)\nabla u)-\nabla \cdot (\psi (u)\nabla v),&(x,t)\in \Omega \times (0,\infty ), \\&0=\Delta v-\mu (t)+f(u),&(x,t)\in \Omega \times (0,\infty ), \end{aligned} \right. \end{aligned}$$

under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset {{\mathbb {R}}}^{n}\), \(n\ge 1\), where \(\mu (t)=\frac{1}{|\Omega |}\int _{\Omega }f(u(x,t))dx\), \(\varphi (u)\) is a nonlinear diffusion, \(\psi (u)\) is a nonlinear sensitivity and f(u) is a nonlinear signal production function. Under suitable assumptions on the functions \(\varphi \), \(\psi \) and f, the global existence and finite-time blow-up of solutions for the above system are studied. The results partially generalize some recent ones obtained in Winkler (Nonlinearity 31:2031–2056, 2018), Li (J Math Anal Appl 480:123376, 2019).

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Acknowledgements

The author would like to deeply thank the editors and reviewers for their insightful and constructive comments. This work is partially supported by National Natural Science Foundation of China (Grant Nos. 11601053, 11526042), Natural Science Foundation of Chongqing (Grant No. cstc2019jcyj-msxmX0082), China-South Africa Young Scientist Exchange Programme 2020, and The Hong Kong Scholars Program (Grant Nos. XJ2021042, 2021-005).

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  1. College of Mathematics and Statistics, Yunnan University, Kunming, 650091, People’s Republic of China

    Pan Zheng

  2. College of Science, Chongqing University of Posts and Telecommunications, Chongqing, 400065, People’s Republic of China

    Pan Zheng

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  1. Pan Zheng
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Correspondence to Pan Zheng.

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Communicated by Joachim Escher.

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Zheng, P. On a generalized volume-filling chemotaxis system with nonlinear signal production. Monatsh Math 198, 211–231 (2022). https://doi.org/10.1007/s00605-022-01669-2

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