Asymptotic behavior of coupled linear systems modeling suspension bridges

We consider the coupled linear system ∂ttu+∂xxxxu+γ∂tu+k(u-v)+h∂t(u-v)=0ϵ∂ttv-∂xxv-k(u-v)-h∂t(u-v)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{\begin{array}{ll} \partial_{tt} u + \partial_{xxxx}u + \gamma \partial_t u + k(u - v) + h\partial_{t} (u-v) = 0\\\epsilon \partial_{tt} v - \partial_{xx}v - k(u - v) - h\partial_{t} (u - v) = 0\end{array}\right.$$\end{document}describing the vibrations of a string-beam system related to the well-known Lazer–McKenna suspension bridge model. For ε > 0 and k > 0, the decay properties of the solution semigroup are discussed in dependence of the nonnegative parameters γ and h, which are responsible for the damping effects.