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Wave propagation and its stability for a class of discrete diffusion systems

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Abstract

This paper is devoted to investigating the wave propagation and its stability for a class of two-component discrete diffusive systems. We first establish the existence of positive monotone monostable traveling wave fronts. Then, applying the techniques of weighted energy method and the comparison principle, we show that all solutions of the Cauchy problem for the discrete diffusive systems converge exponentially to the traveling wave fronts when the initial perturbations around the wave fronts lie in a suitable weighted Sobolev space. Our main results can be extended to more general discrete diffusive systems. We also apply them to the discrete epidemic model with the Holling-II-type and Richer-type effects.

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Acknowledgements

We are very grateful to two anonymous referees for their careful reading and helpful suggestions which led to an improvement of our original manuscript. In addition, the authors would like to thank Mr. Yuji Wan for his contribution in preparing the manuscript, while parallel results can be found in his master thesis [30]. Zhixian Yu is partially supported by the Natural Science Foundation of China (No. 12071297), Science and Technology Innovation Plan of Shanghai (No. 20JC1414200), and Natural Science Foundation of Shanghai (No. 18ZR1426500). Cheng-Hsiung Hsu is partially supported by the MOST and NCTS of Taiwan.

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  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, People’s Republic of China

    Zhixian Yu

  2. Department of Mathematics, National Central University, Zhongli District, Taoyuan City, 32001, Taiwan

    Cheng-Hsiung Hsu

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  1. Zhixian Yu
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  2. Cheng-Hsiung Hsu
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Correspondence to Zhixian Yu.

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Yu, Z., Hsu, CH. Wave propagation and its stability for a class of discrete diffusion systems. Z. Angew. Math. Phys. 71, 194 (2020). https://doi.org/10.1007/s00033-020-01423-4

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  • DOI: https://doi.org/10.1007/s00033-020-01423-4

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