The Cauchy problem and continuation of periodic solutions for a generalized Camassa–Holm equation

ABSTRACT

We consider a three-parameter family of non-linear equations with $(p+1)$-order non-linearities. Such a family includes as a particular member the well-known b-equation, which encloses the famous Camassa–Holm equation. For certain choices of the parameters, we establish a global existence result and show a scenario that prevents the wave breaking of solutions. Also, we explore unique continuation properties for some values of the parameters.

KEYWORDS:

• Camassa–Holm-type equations
• global existence of solution
• continuation of solutions

2020 AMS MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35A01
• 35B60

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

I. L. Freire is grateful to FAPESP for financial support [grant no. 2020/02055-0].