  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.
