Blow-up phenomena for a nonlinear shallow water wave model

We investigate a nonlinear shallow water wave equation, which contains the Fornberg-Whitham model. The L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L<^>{2}$\end{document} conservation law of solutions for the nonlinear equation is derived. Employing the Riccati-type differential inequalities, we give sufficient conditions such that the wave breaking of solutions happens at finite time. The wave-breaking rate and lifespan of solutions are also discussed.