Well-posedness and Blow-up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation

Abstract

We establish local well-posedness in the Sobolev space Hs with any s>32 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the Camassa–Holm equation. However, unlike the more familiar Korteweg–deVries model, we demonstrate conditions on the initial data that lead to finite time blow-up of certain solutions.