Asymptotic behavior of coupled linear systems modeling suspension bridges

We consider the coupled linear system $$ \begin{cases} \partial_{tt} u+\partial_{xxxx}u +\gamma \partial_t u + k(u-v) + h\partial_{t} (u-v)=0\\ \noalign{\vskip1mm} \displaystyle \eps \partial_{tt} v - \partial_{xx}v - k(u-v) - h\partial_{t} (u-v)=0 \end{cases} $$ 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.