Local-in-space wave breaking criteria for a generalized rod equation

Abstract

A generalized rod equation including the celebrated Camassa–Holm shallow water model is considered. The blow-up features of the equation are discussed on the line $R$. Using the method to construct the Lyapunov functions, local-in-space wave breaking criteria to the equation are established. Our wave breaking results are only involved in the initial values $V_0\left(x_0\right)$ and $V_0^{\prime }\left(x_0\right)$ at a single local point $x_0\in R$.