Nonuniform dependence and well-posedness for the generalized Camassa–Holm equation

ABSTRACT

In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation. Using a Galerkin-type approximation scheme, it is shown that this equation is well posed in Sobolev spaces $H^s,s>5/2$ for both the periodic and the nonperiodic case in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates.

KEYWORDS:

• Besov spaces
• Camassa–Holm type equation
• local well-posedness

AMS SUBJECT CLASSIFICATIONS:

• 35B30
• 35G25
• 35A10
• 35Q53

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by NSF of China [grant numbers 11671055, 11771062].