Curvature Blow-up for the Higher-Order Camassa–Holm Equations

  • Changzheng Qu
  • Ying FuEmail author


This paper is devoted to understanding how higher-order nonlinearities affect the dispersive dynamics. As a prototype, a class of higher-order Camassa–Holm equations which can be viewed as a generalization of the Camassa–Holm equation is studied. The local well-posedness of the Cauchy problem in Besov spaces and Sobolev spaces is established. Furthermore, a delicate analysis is employed to investigate the formation of singularities, and some sufficient conditions on initial data that lead to the finite time blow-up of the second-order derivative of the solutions are provided.


Higher-order Camassa–Holm equation Peaked solitary wave Curvature blow-up Local well-posedness Wave-breaking 

Mathematics Subject Classification.

Primary: 35B30 35G25 70H07 



The work of Qu is supported by the National Science Foundation of China grant-11631007 and grant-11971251. The work of Fu is supported by the National Science Foundation of China grant-11471259 and grant-11631007, and the National Science Basic Research Program of Shaanxi (Program No. 2019JM-007).


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Authors and Affiliations

  1. 1.Department of Mathematics, Center for Nonlinear StudiesNingbo UniversityNingboPeople’s Republic of China
  2. 2.School of MathematicsNorthwest UniversityXi’anPeople’s Republic of China

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