The Cauchy problem for the integrable Novikov equation

In this paper we consider the Cauchy problem for the integrable Novikov equation. By using the Littlewood–Paley decomposition and nonhomogeneous Besov spaces, we prove that the Cauchy problem for the integrable Novikov equation is locally well-posed in the Besov space with 1⩽p,r⩽+∞ and . In particular, when with 1⩽p,r⩽+∞ and , for all t∈[0,T], we have that ‖u(t)‖H1=‖u0‖H1. We also prove that the local well-posedness of the Cauchy problem for the Novikov equation fails in .