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 stiffnessk2. When the ends are pinned, long-term dynamics is scrutinized for arbitrary values of axial loadpand stiffnessk2. For a general external sourcef, we prove the existence of bounded absorbing sets. Whenfis 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.
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