On long-time behavior for solutions of the Gear–Grimshaw system

We consider the long-time behavior of the solutions to the Gear–Grimshaw system, which is a type of coupled KdV–KdV systems. We prove two energy decay results to all the solutions of the Gear–Grimshaw systems when the initial datum is in L2(R)×L2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}({\mathbb {R}}) \times L^{2}({\mathbb {R}})$$\end{document} and H1(R)×H1(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{1}({\mathbb {R}}) \times H^{1}({\mathbb {R}})$$\end{document}, respectively. Our results imply that some Gear–Grimshaw systems do not admit time periodic solutions for any initial data in L2(R)×L2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}({\mathbb {R}}) \times L^{2}({\mathbb {R}})$$\end{document} without smallness assumptions.