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 \(\beta \in (0,1/2+\epsilon )\) $\beta \in (0,1/2+ϵ)$ of the structural term, for a small \(\epsilon >0\) $ϵ>0$ , the corresponding problem has a compact global attractor \({\mathfrak {A}}_{\beta }\) ${\mathfrak{A}}_{\beta }$ , which coincides with the unstable manifold \({\mathcal {M}}_{\beta }({\mathcal {N}})\) ${\mathcal{M}}_{\beta }(\mathcal{N})$ emanating from the set \({\mathcal {N}}\) $\mathcal{N}$ 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).