On the curvature blow-up phenomena for the Fokas-Qiao-Xia-Li equation

We study the blow-up phenomena of the strong solutions for the Fokas-Qiao-Xia-Li (FQXL) equation with quadratic and cubic nonlinearities, which include the celebrated Camassa-Holm equation and the Fokas-Olver-Rosenau-Qiao (FORQ) equation as its special case. The FQXL equation models the unidirectional propagation of the shallow water waves over a flat bottom and describes the physics more accurately than its celebrated counterpart, the Gardner equation. Two kinds of conditions on the initial data, including cases where the initial momentum density is not of one sign, are formulated to guarantee finite time blow-up for the strong solutions for this equation. The first condition relies on the sign persistence property of the momentum density m. The second condition is based on the full use of the conservation laws of the equation, especially the conservation quantity H2[u0].