Long-Term Damped Dynamics of the Extensible Suspension Bridge

Abstract EN

This work is focused on the doubly nonlinear equation ∂ttu+∂xxxxu+(p-∥∂xu∥L2(0,1)2)∂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 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.