A note on blow-up criterion of solutions to a Fornberg-Whitham-type equation

In this note, we revisit blow-up phenomena of solutions to a Fornberg-Whitham-type equation, which models shallow-water wave dynamics. We derive a blow-up criterion for the initial data by means of a subtle analysis of a coupled Riccati-type inequality system involving the solution u and its gradient u x . The non-homogeneous terms in this system are functions of time t , and this work extends the previous related results to broader parameter regimes. This method is of independent interest and can be applied to some models whose solutions exhibit growth in time.