Study of the presence and staility global attractors in Riemannian wave equations with localized damping

Abstract

This work addresses the existence and continuity of global attractors for wave equations in Riemannian manifolds, considering the effect of localized damping. Wave equations with localized dissipation represent a relevant model in physical problems, such as wave propagation in media with partial dampers. We have the following questions: Are there exponential global attractors for this type of systems? Is it possible to ensure the continuity of these attractors in the face of external disturbances in the system? The methodology of functional analysis techniques and semigroup theory was used and the existence of a global attractor compact global attractor A in the Hilbert space H. The system, due to the energy dissipation generated by localized damping, meets the necessary conditions of compactness and invariance. It is verified that the energy of the system decreases exponentially: As a result, the trajectories of the system converge asymptotically towards A It is shown that the global attractor Aε associated with a system perturbed by a small variation ε in which the damping coefficient aε(x) converges to the original attractor A. The Hausdorff metric dH between Aε and A satisfies: dH(Aε,A) → 0 cuando ε → 0. Keywords: Riemannian wave equations, Global Attractors, ε-sets controllable in measure.