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Global Solvability in a Two-Species Chemotaxis System with Signal Production

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Abstract

In this paper, we study the following prototypical two-species chemotaxis system with Lotka-Volterra competition and signal production:

$$ \left \{ \textstyle\begin{array}{l@{\quad }l} u_{t}=\Delta u-\nabla \cdot (u\nabla w)+u(1-u^{\theta -1}-v), &x\in \Omega , t>0, \\ v_{t}=\Delta v-\nabla \cdot (v\nabla w)+v(1-v-u), & x\in \Omega , t>0, \\ w_{t}=\Delta w-w+ u+ v, & x\in \Omega , t>0. \end{array}\displaystyle \right .(\ast ) $$

We show that if \(\theta >1+\frac{N-2}{N}\), the associated Neumann initial-boundary value problem for (∗) admits a global generalized solutions in a bounded and smooth domain \(\Omega \subset \mathbb{R}^{N}\) \((N\geq 2)\) under merely integrable initial data.

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Acknowledgements

We thank Prof. Michael Winkler for sharing his preprint [39], we also thank the two anonymous referees for carefully reading our manuscript and giving positive, valuable comments and suggestions, which further helped us to improve the exposition of this work. G. Ren was supported by the National Natural Science Foundation of China (No.12001214). T. Xiang was funded by the National Natural Science Foundation of China (Nos. 12071476 and 11871226).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, P.R. China

    Guoqiang Ren

  2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, P.R. China

    Guoqiang Ren

  3. Institute for Mathematical Sciences, Renmin University of China, Bejing, 100872, P.R. China

    Tian Xiang

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  1. Guoqiang Ren
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Correspondence to Tian Xiang.

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Ren, G., Xiang, T. Global Solvability in a Two-Species Chemotaxis System with Signal Production. Acta Appl Math 178, 12 (2022). https://doi.org/10.1007/s10440-022-00485-y

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