General energy decay rates for a weakly damped Timoshenko system

Abstract

In this paper, we consider the following Timoshenko-type system:

$$ \left\{ {\begin{array}{*{20}{c}} {{\varphi_{tt}} - {{\left( {{\varphi_x} + \psi } \right)}_x} = 0} \hfill & {{\text{in }}\left( {0,1} \right) \times {\mathbb{R}_{+} },} \hfill \\ {{\psi_{tt}} - {\psi_{xx}} + {\varphi_x} + \psi + \alpha (t)g{{\left( {{\psi_t}} \right)}} = 0} \hfill & {{\text{in }}\left( {0,1} \right) \times {\mathbb{R}_{+} }.} \hfill \\ \end{array} } \right. $$

We establish an explicit and general decay result, depending on g and α, using some properties of convex functions. We obtain our result without imposing any restrictive growth assumption on g at the origin.