Abstract
In this paper, the formation of singularities for the nonlocal Whitham-type equations is studied. It is shown that if the lowest slope of flows can be controlled by its highest value with the bounded Whitham-type integral kernel initially, then the finite-time blow-up will occur in the form of wave-breaking. This refined wave-breaking result is established by analyzing the monotonicity and continuity properties of a new system of the Riccati-type differential inequalities involving the extremal slopes of flows. Our theory is illustrated via the Whitham equation, Camassa–Holm equation, Degasperis–Procesi equation, and their μ-versions as well as hyperelastic rod equation.
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