# American Institute of Mathematical Sciences

November  2016, 15(6): 2301-2328. doi: 10.3934/cpaa.2016038

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

 1 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588 2 Department of Mathematics, Faculty of Science, Hacettepe University, Beytepe 06800, Ankara 3 Haceteppe University, Ankara , Turkey

Received  February 2016 Revised  July 2016 Published  September 2016

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 $(\Delta u)\big|_{\Gamma_0}$ 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).
Citation: George Avalos, Pelin G. Geredeli, Justin T. Webster. Finite dimensional smooth attractor for the Berger plate with dissipation acting on a portion of the boundary. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2301-2328. doi: 10.3934/cpaa.2016038
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