Wave breaking for the periodic generalized Camassa-Holm equation with dual-power nonlinearities

Considered herein is the blow-up mechanism to the periodic generalized Camassa-Holm equations with dual-power nonlinearities. We first establish the local well-posedness by applying the Kato's semigroup theory.The precise blow-up scenarios of strong solutions and several sufficient conditions on the initial data to guarantee blow-up of the induced solutions are described in detail according to the different real-valued intervals in which the dispersive parameter s is located. The challenge we need to meet is to readjust the corresponding spatial estimates in order to balance the relationship between the various dispersive parameters and overcome the difficulty caused by complicated mixed dual-power nonlinear structure like the non-periodic case.