Attractors for a class of wave equations with nonlocal structural energy damping

This paper is concerned with the well-posedness and long-time dynamics for a class of wave equations with a nonlocal structural energy damping. The main result establishes that for each exponent β ∈ ( 0 , 1 / 2 + ϵ ) of the structural term, for a small ϵ > 0 , the corresponding problem has a compact global attractor A β , which coincides with the unstable manifold M β ( N ) emanating from the set N of stationary points. This class of problems whose dissipation intensity depends on the energy of the system has connection with flight structure models, see NASA-AirForce reports (Balakrishnan in A theory of nonlinear damping in flexible structures. Stabilization of flexible structures, 1988; Balakrishnan and Taylor in Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989).