Curvature Blow-up for the Higher-Order Camassa-Holm Equations

This paper is devoted to understanding how higher-order nonlinearities affect the dispersive dynamics. As a prototype, a class of higher-order Camassa-Holm equations which can be viewed as a generalization of the Camassa-Holm equation is studied. The local well-posedness of the Cauchy problem in Besov spaces and Sobolev spaces is established. Furthermore, a delicate analysis is employed to investigate the formation of singularities, and some sufficient conditions on initial data that lead to the finite time blow-up of the second-order derivative of the solutions are provided.