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Boundedness in a three-dimensional two-species chemotaxis system with two chemicals

Abstract

This paper deals with a two-species chemotaxis system with Lotka–Volterra competitive kinetics

$$\begin{aligned} \left\{ \begin{array}{llll} u_t=\Delta u-\chi _1\nabla \cdot (u\nabla v)+\mu _1u(1-u-a_1w),&{}\quad x\in \Omega ,\quad t>0,\\ v_t=\Delta v-v+w,&{}\quad x\in \Omega ,\quad t>0,\\ w_t=\Delta w-\chi _2\nabla \cdot (w\nabla z)+\mu _2w(1-w-a_2u),&{}\quad x\in \Omega ,\quad t>0,\\ z_t=\Delta z-z+u,&{}\quad x\in \Omega ,\quad t>0, \end{array} \right. \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^3\) is a bounded domain with smooth boundary \(\partial \Omega \); the parameters \(\chi _i,\,\,\mu _i\,\,\text {and}\,\,a_i\) \((i = 1,2)\) are positive, and the nonnegative initial data \((u_{0}, v_{0},w_{0}, z_{0})\in C^{0}({\overline{\Omega }})\times W^{1,\infty }(\Omega )\times C^{0}({\overline{\Omega }})\times W^{1, \infty }(\Omega )\). It is shown that the system possesses a unique global bounded classical solution provided that \(\mu _i (i=1,2)\) are sufficiently large.

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Acknowledgements

The authors thank the anonymous referee for his/her positive and useful comments, which helped him improve further the exposition of the paper. This work is supported by the National Natural Science Foundation of China (No. 11601052) and Chongqing Basic Science and Advanced Technology Research Program (Nos. cstc2017jcyjAX0178 and cstc2017jcyjXB0037).

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Correspondence to Liangchen Wang.

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Pan, X., Wang, L., Zhang, J. et al. Boundedness in a three-dimensional two-species chemotaxis system with two chemicals. Z. Angew. Math. Phys. 71, 26 (2020). https://doi.org/10.1007/s00033-020-1248-2

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Mathematics Subject Classification

  • 35K35
  • 35A01
  • 35B44
  • 35B35
  • 92C17

Keywords

  • Boundedness
  • Lotka–Volterra competition
  • Two-species chemotaxis