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Existence of Traveling Waves of General Gray-Scott Models

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Abstract

This work gives a rigorous proof of the existence of propagating traveling waves of a nonlinear reaction–diffusion system which is a general Gray-Scott model of the pre-mixed isothermal autocatalytic chemical reaction of order m (\(m > 1\)) between two chemical species, a reactant A and an auto-catalyst B, \( A + m B \rightarrow (m+1) B\), and a super-linear decay of order \( n > 1\), \( B \rightarrow C\), where \( 1< n < m\). Here C is an inert product. Moreover, we establish that the speed set for existence must lie in a bounded interval for a given initial value \(u_0\) at \( - \infty \). The explicit bound is also derived in terms of \(u_0\) and other parameters. The same system also appears in a mathematical model of SIR type in infectious diseases.

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Author information

Authors and Affiliations

  1. School of Mathematics, Peking University, Bejing, 100871, China

    Zhi Zheng

  2. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Xinfu Chen

  3. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Yuanwei Qi

  4. School of Mathematical Sciences, Peking University, Bejing, 100871, China

    Shulin Zhou

Authors
  1. Zhi Zheng
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  2. Xinfu Chen
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  3. Yuanwei Qi
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  4. Shulin Zhou
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Corresponding author

Correspondence to Yuanwei Qi.

Additional information

Zheng is supported by China Scholarship Council. Chen is partially supported by NSF Grant DMS-1008905 and hundred experts program of Shanxi province. Zhou was partially supported by by the NSFC under Grant 11571020. The authors thank Xuefeng Wang and Xiaoqiang Zhao for stimulating discussions.

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Zheng, Z., Chen, X., Qi, Y. et al. Existence of Traveling Waves of General Gray-Scott Models. J Dyn Diff Equat 30, 1469–1487 (2018). https://doi.org/10.1007/s10884-017-9603-5

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  • DOI: https://doi.org/10.1007/s10884-017-9603-5

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