Blow-up Analysis for the ab -Family of Equations

This paper investigates the Cauchy problem for the ab -family of equations with cubic nonlinearities, which contains the integrable modified Camassa–Holm equation ( a = 1/3 , b = 2 ) and the Novikov equation ( a = 0 , b = 3 ) as two special cases. When 3a + b 3 , the ab -family of equations does not possess the H^1 -norm conservation law. We give the local well-posedness results of this Cauchy problem in Besov spaces and Sobolev spaces. Furthermore, we provide a blow-up criterion, the precise blow-up scenario and a sufficient condition on the initial data for the blow-up of strong solutions to the ab -family of equations. Our blow-up analysis does not rely on the use of the conservation laws.