Blow-up Analysis for the \({\varvec{ab}}\)-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 ( \(a = \frac{1}{3}\) $a=\frac{1}{3}$ , \(b = 2\) $b=2$ ) and the Novikov equation ( \(a = 0\) $a=0$ , \(b = 3\) $b=3$ ) as two special cases. When \(3a + b \ne 3\) $3a+b\ne 3$ , the ab-family of equations does not possess the \(H^1\) $H^1$ -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.