Blow-up scenario for a generalized Camassa–Holm equation with both quadratic and cubic nonlinearity

ABSTRACT

In this paper, we mainly devote to study an integrable generalized Camassa–Holm equation with both quadratic and cubic nonlinearity proposed by Xia, Qiao and Li [An integrable system with peakon, complex peakon, weak kink, and kink-peakon interactional solutions. Commun Nonlinear Sci Numer Simul. 2018;63:292–306]. By overcoming the difficulties cased by the complicated mixed nonlinear structure, we present a precise blow-up scenario and some new blow-up results for strong solutions under different initial data.

KEYWORDS:

• Generalized Camassa–Holm equation
• integrable equation
• blow-up criteria

MATHEMATICS SUBJECT CLASSIFICATIONS (2010):

• 35A01
• 35B44
• 35B65
• 35G25

Acknowledgments

The author would like to thank the editor and the referees for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China (grant nos. 11671033, A010802).