On the Cauchy problem for a modified Camassa–Holm equation

Abstract

In this paper, we first study the local well-posedness for the Cauchy problem of a modified Camassa–Holm equation in nonhomogeneous Besov spaces. Then we obtain a blow-up criteria and present a blow-up result for the equation. Finally, with proving the norm inflation we show the ill-posedness occurs to the equation in critical Besov spaces.

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Acknowledgements

This work was partially supported by NNSFC (No. 11671407), FDCT (No. 0091/2018/A3), Guangdong Special Support Program (No. 8-2015), and the key project of NSF of Guangdong province (No. 2016A03031104). The author Qiao thanks the UT President Endowed Professorship (Project # 450000123) for its partial support.

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Correspondence to Zhaoyang Yin.

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Communicated by Adrian Constantin.

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Luo, Z., Qiao, Z. & Yin, Z. On the Cauchy problem for a modified Camassa–Holm equation. Monatsh Math 193, 857–877 (2020). https://doi.org/10.1007/s00605-020-01426-3

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Keywords

  • A modified Camassa–Holm equation
  • Bseov spaces
  • Local well-posedness
  • Blow up
  • Ill-posedness

Mathematics Subject Classification

  • 35Q53
  • 35A01
  • 35B44
  • 35B65