Curvature blow-up for the higher-order Camassa-Holm equations (Q2211291)

This paper deals with the Camassa-Holm-type equations with higher-order nonlinearity \[ m_t+\Big(u^2-(u_x)^2\Big)^q u_xm+\Big(\Big(u^2-(u_x)^2\Big)^q um\Big)=0, \quad m = u - u_{xx},\quad q\ge0. \] The authors study the local well-posedness of the Cauchy problem in Besov spaces and Sobolev spaces, the formation of singularities, and sufficient conditions on initial data that lead to the finite time blow-up of the second-order derivative of the solutions. One of the key ingredients in their analysis is the preservation of the sign of \(m\) along the characteristics.