Global existence and propagation speed for a generalized Camassa–Holm model with both dissipation and dispersion

Abstract

In this paper, we study a generalized Camassa–Holm (gCH) model with both dissipation and dispersion, which has (\(N+1\))-order nonlinearities and includes the following three integrable equations: the Camassa–Holm, the Degasperis–Procesi, and the Novikov equations, as its reductions. We first present the local well-posedness and a precise blow-up scenario of the Cauchy problem for the gCH equation. Then, we provide several sufficient conditions that guarantee the global existence of the strong solutions to the gCH equation. Finally, we investigate the propagation speed for the gCH equation when the initial data are compactly supported.

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Acknowledgements

This work was supported by NSF of China (No. 11401223), NSF of Guangdong (No. 2015A030313424), the Science and Technology Program of Guangzhou (Grant No. 201607010005), and China Scholarship Council. The first author would like to thank Professor Zhijun Qiao for his kind hospitality and encouragement during her visit in the University of Texas Rio Grande Valley, and the second author (Qiao) thanks the UT Presidents Endowed Professorship (Project # 450000123) for its partial support.

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Hu, Q., Qiao, Z. Global existence and propagation speed for a generalized Camassa–Holm model with both dissipation and dispersion. J. Evol. Equ. 20, 403–419 (2020). https://doi.org/10.1007/s00028-019-00533-5

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Mathematics Subject Classification

  • 35G25
  • 35L05

Keywords

  • Generalized Camassa–Holm (gCH) model
  • Local well-posedness
  • Blowup
  • Global solution
  • Propagation speed