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Well-posedness of short time solutions and non-uniform dependence on the initial data for a shallow water wave model in critical Besov space

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Abstract

A nonlinear shallow water wave equation containing the famous Degasperis-Procesi and Fornberg-Whitham equations is investigated. The well-posedness of short time solutions is established to illustrate that the solution map of the equation is continuous in the critical Besov space \(B^{1}_{\infty ,1}(\mathbb {R})\). Using the methods to construct high and low frequency functions, we prove that the solution map of the equation is non-uniform continuous dependence on the initial data in \(B^{1}_{\infty ,1}(\mathbb {R})\).

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Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Acknowledgements

Thanks are given to the reviewers for their valuable suggestions, which lead to the meaningful improvement of the paper. This work is supported by National Nature Science Foundation of China (No. 12361042) and 14th Five Years’ Key Discipline of Xinjiang Uygur Autonomous Region (No. 83451378).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Yili Normal University, Yining, 835000, China

    Changtai Zhou & Honglin Xiao

  2. School of Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China

    Shaoyong Lai

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  1. Changtai Zhou
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  2. Honglin Xiao
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Correspondence to Honglin Xiao.

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Communicated by Adrian Constantin.

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Zhou, C., Xiao, H. & Lai, S. Well-posedness of short time solutions and non-uniform dependence on the initial data for a shallow water wave model in critical Besov space. Monatsh Math (2024). https://doi.org/10.1007/s00605-024-01959-x

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