Long-Term Damped Dynamics of the Extensible Suspension Bridge

Abstract

This work is focused on the doubly nonlinear equation ${\partial }_{tt}u+{\partial }_{xxxx}u+(p-\parallel {\partial }_{x}u{\parallel }_{L^2\left(\mathrm{0,1}\right)}^2){\partial }_{xx}u+{\partial }_{t}u+k^2{u}^{+}=f$, whose solutions represent the bending motion of an extensible, elastic bridge suspended by continuously distributed cables which are flexible and elastic with stiffness $k^2$. When the ends are pinned, long-term dynamics is scrutinized for arbitrary values of axial load $p$ and stiffness $k^2$. For a general external source $f$, we prove the existence of bounded absorbing sets. When $f$ is time-independent, the related semigroup of solutions is shown to possess the global attractor of optimal regularity and its characterization is given in terms of the steady states of the problem.