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On boundedness, blow-up and convergence in a two-species and two-stimuli chemotaxis system with/without loop

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Abstract

In this work, we study dynamic properties of classical solutions to a homogenous Neumann initial-boundary value problem (IBVP) for a two-species and two-stimuli chemotaxis model with/without chemical signalling loop in a 2D bounded and smooth domain. We detect the product of two species masses as a feature to determine boundedness, gradient estimate, blow-up and exponential convergence of classical solutions for the corresponding IBVP. More specifically, we first show generally a smallness on the product of both species masses, thus allowing one species mass to be suitably large, is sufficient to guarantee global boundedness, higher order gradient estimates and \(W^{j,\infty }(j\ge 1)\)-exponential convergence with rates of convergence to constant equilibria; and then, in a special case, we detect a straight line of masses on which blow-up occurs for large product of masses. Our findings provide new understandings about the underlying model, and thus, improve and extend greatly the existing knowledge relevant to this model.

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Acknowledgements

The authors thank the anonymous referee very much for carefully reading our manuscript and giving positive and valuable comments, which further helped them to improve the exposition of this work. K. Lin is supported by the NSF of China (No. 11801461), and T. Xiang is supported by the NSF of China (Nos. 11601516 and 11871226) and the Research Funds of Renmin University of China (No. 2018030199).

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Authors and Affiliations

  1. School of Economics and Mathematics, Southwestern University of Economics and Finance, Chengdu, 610074, Sichu, China

    Ke Lin

  2. Institute for Mathematical Sciences, Renmin University of China, Bejing, 100872, China

    Tian Xiang

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  1. Ke Lin
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  2. Tian Xiang
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Correspondence to Tian Xiang.

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Communicated by P. Rabinowitz.

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Lin, K., Xiang, T. On boundedness, blow-up and convergence in a two-species and two-stimuli chemotaxis system with/without loop. Calc. Var. 59, 108 (2020). https://doi.org/10.1007/s00526-020-01777-7

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