Long-Time Dynamics of a Semilinear Beam in a Contact Problem with Pointwise Damping

This work investigates the existence of global attractors for a semilinear Signorini problem associated with the Euler–Bernoulli beam equation featuring pointwise dissipation. We demonstrate that the system exhibits exponential decay to zero and possesses a compact global attractor. The analysis is conducted by approximating the original linearized problem through a family of hybrid PDE–ODE models. By employing Lipschitz perturbations, we establish the well-posedness and global existence of solutions for the semilinear case. Finally, the Signorini problem is recovered via a singular limit process, where it is rigorously proven that the transition from the hybrid model to the constrained Signorini problem preserves both the exponential stability and the topological structure of the global attractor.