  For models describing water waves, Constantin and Escher'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. Even though the proofs of
these involve a system of coupled Riccati-type differential inequalities,
however, little or no attention has been made to a generalization of this
Riccati-type system. 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. Up to the author's knowledge, the blow-up
analysis upon this structural generalization is new and is of theoretical
interest in itself as well as its application to various non-local flow models.
The theory is illustrated via the Whitham-type equation with nonlinear drift.
Our method is applicable to a large class of non-local conservation laws.
