Wave breaking in a class of non-local conservation laws

For models describing water waves, Constantin and Escher [3]'s works have long been considered as the cornerstone method for proving wave breaking phenomena. Their rigorous analytic proof shows that if the lowest slope of flows can be controlled by its highest slope initially, then the wave-breaking occur for the Whitham-type equation. Since this breakthrough, there have been numerous refined wave-breaking results established by generalizing the kernel which describes the dispersion relation of water waves. In this work, from a rich class of non-local conservation laws, a Riccati-type system that governs the flow's gradient is extracted and investigated. The system's leading coefficient functions are allowed to change their values and signs over time as opposed to the ones in many of other previous works are fixed constants. The blow-up analysis is illustrated via the Whitham-type equation with nonlinear drift.