Wave-breaking criteria of solution for a Fornberg-Whitham type equation revisited

Abstract

In this paper, we mainly revisit to a Fornberg-Whitham type equation, which can be derived as a special shallow water wave equation of the Constantin-Lannes-type models proposed by Constantin and Lannes (2009). We focus on some new wave-breaking criteria of the solution for the equation on the line or circle based on the different real-valued intervals in which the dispersive parameter $m$ being located. A prior estimate of $L^{\infty }$-norm of the solution for equation is first obtained by the interval of the dispersive parameter $m$. By this estimate and a weaker conserved $L^2$-norm, we then study some sufficient conditions which guarantee the occurrence of wave-breaking of solutions on the line. It is worthy noting that the results we obtained not only supplement the wave-breaking results of classic FW equation on the line in the previous references, but also extend these results to a wider range of dispersive parameters $k$ and $m$. Moreover, we give the wave-breaking criterion of the solution for equation on the circle without utilizing any conservation law.