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Boundedness and Stability in a Chemotaxis-Growth Model with Indirect Attractant Production and Signal-Dependent Sensitivity

  • Shuyan QiuEmail author
  • Chunlai Mu
  • Yafeng Li
Article
  • 16 Downloads

Abstract

We study the chemotaxis-growth system with signal-dependent sensitivity function and logistic source
$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} u_{t}=\Delta u-\nabla \cdot \bigl(u\chi (v)\nabla v\bigr)+\mu u(1-u), &x \in \varOmega ,\ t>0, \\ v_{t}=d\Delta v+h(v,w), &x\in \varOmega ,\ t>0, \\ \tau w_{t}=-\delta w+u, &x\in \varOmega ,\ t>0, \\ \end{array}\displaystyle \right . \end{aligned}$$
under homogeneous Neumann boundary conditions in a smooth bounded domain \(\varOmega \subset \mathbb{R}^{n}\ (n\geq 1)\), where the parameters \(\mu , \tau , \delta >0\) and \(d\geq 0\), the functions \(\chi (v)\), \(h(v,w)\) satisfying some conditions represent the chemotactic sensitivity and the balance between the production and degradation of the chemical signal which relies explicitly on the living organisms, respectively. In the case that \(\chi (v)\equiv 1\), \(d=1\) and \(h(v,w)=-v+w\), Hu and Tao (Math. Models Methods Appl. Sci. 26:2111–2128, 2016) asserted global existence of bounded solutions for arbitrary \(\mu >0\) and established asymptotic behavior of solutions to the mentioned system under the condition \(\mu >\frac{1}{8\delta ^{2}}\) in the three dimensional space. The purpose of the present paper is to investigate the global existence and boundedness of classical solutions and to improve the condition assumed in Hu and Tao (Math. Models Methods Appl. Sci. 26:2111–2128, 2016) by extending the previous method for obtaining asymptotic stability. Consequently, the range of \(\mu \) is extended in the present paper.

Keywords

Chemotaxis Boundedness Indirect attractant production Logistic growth Asymptotic behavior 

Mathematics Subject Classification

35A01 92C17 35B45 35B40 35K57 35Q92 

Notes

Acknowledgements

The second author is supported by NSFC (Grant Nos. 11571062 and 11771062), the Fundamental Research Funds for the Central Universities (Grant Nos. 106112016CDJXZ238826 and 2019CDJCYJ001).

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsChongqing UniversityChongqingP.R. China

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