Long-Term Damped Dynamics of the Extensible Suspension Bridge - Ivana Bochicchio; Claudio Giorgi; Elena Vuk (International Journal of Differential Equations)
This work is focused on the doubly nonlinear equation 𝜕𝑡𝑡𝑢+𝜕𝑥𝑥𝑥𝑥𝑢+(𝑝−‖𝜕𝑥𝑢‖2𝐿2(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 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.