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International Journal of Differential Equations
Volume 2010, Article ID 383420, 19 pages
http://dx.doi.org/10.1155/2010/383420
Research Article

Long-Term Damped Dynamics of the Extensible Suspension Bridge

1Dipartimento di Matematica e Informatica, Università degli studi di Salerno, 84084 Fisciano, Italy
2INFN, Sezione di Napoli, Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy
3Dipartimento di Matematica, Università degli studi di Brescia, 25133 Brescia, Italy

Received 29 September 2009; Revised 14 December 2009; Accepted 14 January 2010

Academic Editor: Maurizio Grasselli

Copyright © 2010 Ivana Bochicchio et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This work is focused on the doubly nonlinear equation 𝜕 𝑡 𝑡 𝑢 + 𝜕 𝑥 𝑥 𝑥 𝑥 𝑢 + ( 𝑝 𝜕 𝑥 𝑢 2 𝐿 2 ( 0 , 1 ) ) 𝜕 𝑥 𝑥 𝑢 + 𝜕 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.