Weak well-posedness for the integrable modified Camassa-Holm equation with the cubic nonlinearity

Abstract

The goal of this paper is supposed to investigate the weak well-posedness for the modified Camassa-Holm equation with cubic nonlinearity after given an initial data u0∈Hs(R)∩W2,∞(R)(1≤s≤2). Firstly, we deduce an ODE system, whose equivalence to problem (1.2) with a spatial initial condition will be verified later. Next, by studying the existence of the unique solution to the ODE system, we conclude the corresponding results of the original problem. Finally, we end up with the weak continuity of solutions to the PDE associated to the initial datum.