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Global boundedness and asymptotic behavior in an attraction–repulsion chemotaxis system with nonlocal terms

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Abstract

In this paper, we consider the attraction–repulsion chemotaxis system with nonlocal terms

$$\begin{aligned} \left\{ \begin{array}{lll} u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+\xi \nabla \cdot (u\nabla w)+f(u), &{}x\in \Omega , t>0, \\ \tau _1v_t=\Delta v-\alpha _1v+\beta _1u, &{} x\in \Omega , t>0, \\ \tau _2w_t=\Delta w-\alpha _2w+\beta _2u, &{} x\in \Omega , t>0 \end{array}\right. (*) \end{aligned}$$

under Neumann boundary conditions in a bounded domain with smooth boundary, where \(f(u)=u^{\sigma }(a_0-a_1u-a_2 \int \limits _{\Omega }u^{\beta })\). We show that the system (\(*\)) possesses a unique global classical solution in three different cases (parabolic–elliptic–elliptic, fully parabolic, parabolic–parabolic–elliptic). Our results generalize and improve partial previously known ones.

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Acknowledgements

The author would like to express his gratitude to Professor Bin Liu for helpful discussions during the preparation of the paper and moreover express his gratitude to the anonymous reviewers and editors for their valuable comments and suggestions which led to the improvement in the original manuscript. Guoqiang Ren is supported by NSF of China (Grant No 12001214) and the Fundamental Research Funds for the Central Universities (Grant No. 3004011139).

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  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, People’s Republic of China

    Guoqiang Ren

  2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, People’s Republic of China

    Guoqiang Ren

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Ren, G. Global boundedness and asymptotic behavior in an attraction–repulsion chemotaxis system with nonlocal terms. Z. Angew. Math. Phys. 73, 200 (2022). https://doi.org/10.1007/s00033-022-01832-7

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