Long-Time Behaviour for Solutions of Systems of PDEs Modelling Suspension Bridges

We address a damped and forced system of PDEs modelling the dynamics of a degenerate plate with internal piers, where the two unknowns represent the vertical displacement of the central beam and the angular displacement of the cross sections with respect to the horizontal plane. After discussing the well-posedness of the problem, we prove the boundedness of the solutions in the presence of bounded forcing terms and positive damping on both the components. We then highlight the difficulties arising when the damping is concentrated on the longitudinal component only, showing that in this case the energy is bounded by a quadratic function of time. The exploited technique relies on direct estimates of suitable energy functionals associated with the considered problem. The results are complemented by some numerical experiments.