Global existence for the periodic dispersive Hunter–Saxton equation

In this paper, we study an integrable dispersive Hunter–Saxton equation in periodic domain. Firstly, we establish the local well-posedness of the Cauchy problem of the equation in Hs(S),s≥2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^s ({\mathbb {S}}), s \ge 2,$$\end{document} by applying the Kato method. Then, based on a sign-preserve property, we obtain a global existence result for the equation. Moreover, we extend the obtained result to some periodic nonlinear partial differential equations of second order of the general form.