  Considered herein is the initial-value problem for the generalized periodic
Camassa-Holm equation which is related to the Camassa-Holm equation and the
Hunter-Saxton equation. Sufficient conditions guaranteeing the development of
breaking waves in finite time are demonstrated. On the other hand, the
existence of strong permanent waves is established with certain initial
profiles depending on the linear dispersive parameter in a range of the Sobolev
spaces. Moreover, the admissible global weak solution in the energy space is
obtained.
