The semilinear suspension bridge model with pointwise dissipation
This paper is devoted to the analysis of a semilinear suspension bridge model with pointwise localized dissipation. The main contribution of the work is the development of a robust semigroup framework that substantially simplifies the stability analysis of coupled systems with localized damping. The core of the approach relies on a comparison principle involving the essential growth bound of the associated semigroups together with the invariance of the essential spectrum under compact perturbations. This methodology allows us to reduce the stability analysis of the full coupled system to that of suitably chosen auxiliary systems, thereby avoiding the delicate multiplier techniques and observability arguments commonly used in the literature. As a first consequence, we establish the exponential stability of the corresponding linear homogeneous system. The semilinear problem is then treated as a locally Lipschitz perturbation of the exponentially stable linear semigroup, yielding global well-posedness and exponential decay of solutions. Finally, we investigate the long-time dynamics of the semilinear system. By combining the dissipative structure of the model with suitable regularization properties of the nonlinear semigroup, we prove the existence of a global compact attractor and an exponential attractor with finite fractal dimension. These results provide a complete description of the asymptotic behavior of solutions and show that the proposed framework is sufficiently flexible to be adapted to other coupled systems with localized dissipation.