General energy decay rates for a weakly damped Timoshenko system

In this paper, we consider the following Timoshenko-type system: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{ {\begin{array}{*{20}{c}} {{\varphi_{tt}} - {{\left( {{\varphi_x} + \psi } \right)}_x} = 0} \hfill & {{\text{in }}\left( {0,1} \right) \times {\mathbb{R}_{+} },} \hfill \\ {{\psi_{tt}} - {\psi_{xx}} + {\varphi_x} + \psi + \alpha (t)g{{\left( {{\psi_t}} \right)}} = 0} \hfill & {{\text{in }}\left( {0,1} \right) \times {\mathbb{R}_{+} }.} \hfill \\ \end{array} } \right. $$\end{document}