Finite dimensional smooth attractor for the Berger plate with dissipation acting on a portion of the boundary

Abstract

We consider a (nonlinear) Berger plate in the absence of rotational inertia acted upon by nonlinear boundary dissipation. We take the boundary to have two disjoint components: a clamped (inactive) portion and a controlled portion where the feedback is active via a hinged-type condition. We emphasize the damping acts only in one boundary condition on a portion of the boundary. In [24] this type of boundary damping was considered for a Berger plate on the whole boundary and shown to yield the existence of a compact global attractor. In this work we address the issues arising from damping active only on a portion of the boundary, including deriving a necessary trace estimate for (Δμ)|ᵣ₀ and eliminating a geometric condition in [24] which was utilized on the damped portion of the boundary. Additionally, we use recent techniques in the asymptotic behavior of hyperbolic-like dynamical systems [11, 18]involving a ``stabilizability' estimate to show that the compact global attractor has finite fractal dimension and exhibits additional regularity beyond that of the state space (for finite energy solutions).