Wave-breaking phenomena for a weakly dissipative shallow water equation

Consideration in the present paper is a weakly dissipative shallow water equation. The parameters take different values, which include several different important shallow water equations, such as CH equation, DP equation, Novikov equation and so on. The wave-breaking phenomena are investigated by three different kinds of method. Due to the presence of high-order nonlinear terms u2n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{2n+1}$$\end{document} and u2muxx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{2m}u_{xx}$$\end{document}, the equation loses the conservation law E=∫S(u2+ux2)dx.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E=\int _{{\mathbb {S}}} (u^2+u^2_x)\mathrm{d}x.$$\end{document} This difficulty has been dealt with by establishing the energy inequality.