Long-Term Damped Dynamics of the Extensible Suspension Bridge

This work is focused on the doubly nonlinear equation ∂ttu ∂xxxxu p − ‖∂xu‖L2 0,1 ∂xxu ∂tu k2u f , whose solutions represent the bending motion of an extensible, elastic bridge suspended by continuously distributed cables which are flexible and elastic with stiffness k2. When the ends are pinned, long-term dynamics is scrutinized for arbitrary values of axial load p and stiffness k2. For a general external source f , we prove the existence of bounded absorbing sets. When f is timeindependent, 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.