Blow-up Analysis for the $${\varvec{ab}}$$ -Family of Equations

Abstract

This paper investigates the Cauchy problem for the ab-family of equations with cubic nonlinearities, which contains the integrable modified Camassa–Holm equation (\(a = \frac{1}{3}\), \(b = 2\)) and the Novikov equation (\(a = 0\), \(b = 3\)) as two special cases. When \(3a + b \ne 3\), 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.