Exponential decay rate of a nonlinear suspension bridge model by a local distributed and boundary dampings

In this paper, we investigate the decay properties of the unconstrained one dimensional suspension bridge model. With only partial damping acting on one or on both equations and with boundary dampings, we prove that the first order energy is decaying exponentially, our method of proof is based on the energy method to build the appropriate Lyapunov functional. Moreover, we develop a numerical algorithm which is based on the finite element method to approximate the spatial variable and the Crank–Nicolson type of symmetric difference scheme to discretize the time derivative, and also a Picard type iteration process for solving the system of nonlinear equations obtained by discretization. At the end, we present some numerical experiments to illustrate our theoretical results.