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Blow-up phenomena and local well-posedness for a generalized Camassa–Holm equation in the critical Besov space

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Abstract

In this paper we mainly study the Cauchy problem for a generalized Camassa–Holm equation in a critical Besov space. First, by using the Littlewood–Paley decomposition, transport equations theory, logarithmic interpolation inequalities and Osgood’s lemma, we establish the local well-posedness for the Cauchy problem of the equation in the critical Besov space \(B^{\frac{1}{2}}_{2,1}\). Next we derive a new blow-up criterion for strong solutions to the equation. Then we give a global existence result for strong solutions to the equation. Finally, we present two new blow-up results and the exact blow-up rate for strong solutions to the equation by making use of the conservation law and the obtained blow-up criterion.

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Acknowledgements

This work was partially supported by NNSFC (No. 11671407 and No. 11801076), FDCT (No. 0091/2018/A3), Guangdong Special Support Program (No. 8-2015) and the key project of NSF of Guangdong Province (No. 2016A030311004). The authors thank the referee for valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Foshan University, Foshan, 528000, China

    Xi Tu

  2. Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

    Zhaoyang Yin

  3. Faculty of Information Technology, Macau University of Science and Technology, Macau, China

    Zhaoyang Yin

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  1. Xi Tu
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  2. Zhaoyang Yin
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Correspondence to Zhaoyang Yin.

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Communicated by Joachim Escher.

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Tu, X., Yin, Z. Blow-up phenomena and local well-posedness for a generalized Camassa–Holm equation in the critical Besov space. Monatsh Math 191, 801–829 (2020). https://doi.org/10.1007/s00605-020-01371-1

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  • DOI: https://doi.org/10.1007/s00605-020-01371-1

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