Global existence and wave-breaking criteria for the nonlocal Whitham-type equations

In this paper, we investigate some nonlocal Whitham-type equations which include the nonlinear dispersive θ -equation, the Fornberg-Whitham-Degasperis-Procesi equations and the generalized rod equation. A class of priori estimates of L ∞ -norm of the solution to the corresponding equations is firstly constructed. Then, a novel global existence and two wave-breaking criteria involving different range of the parameter θ to the nonlinear dispersive θ -equation are obtained without relying on any conservation law properties. Subsequently, comparing with the results of previous references Shu (2025) [54] , we consider some new wave-breaking criteria of the Fornberg-Whitham-Degasperis-Procesi equations depending on the piecewise intervals of the parameter b a . Finally, we obtain the global existence for the generalized rod equation with the parameters β γ 0 ∈ [ 2 , 3 ] . Some wave-breaking criteria of the generalized rod equation are studied by discussing the specific range of β γ 0 , which refines and extends the results from previous references Zhou and Lai (2024) [64] .