On the blow-up structure for the generalized periodic Camassa–Holm and Degasperis–Procesi equations ✩

Considered herein are the generalized Camassa–Holm and Degasperis–Procesi equations in the spatially periodic setting. The precise blow-up scenarios of strong solutions are derived for both of equations. Several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established. The exact blow-up rates are also determined. Finally, geometric descriptions of these two integrable equations from non-stretching invariant curve flows in centro-equiaffine geometries, pseudo-spherical surfaces and affine surfaces are given. © 2012 Elsevier Inc. All rights reserved.