Communications on Pure and Applied Analysis (CPAA)

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

Pages: 2301 - 2328, Volume 15, Issue 6, November 2016      doi:10.3934/cpaa.2016038

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George Avalos - Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588, United States (email)
Pelin G. Geredeli - Department of Mathematics, Faculty of Science, Hacettepe University, Beytepe 06800, Ankara, Turkey (email)
Justin T. Webster - Haceteppe University, Ankara , Turkey (email)

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 $(\Delta u)\big|_{\Gamma_0}$ 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).

Keywords:  Global attractor, nonlinear plate equation, boundary dissipation, dissipative dynamical system.
Mathematics Subject Classification:  Primary: 35B41, 74K20; Secondary: 35Q74, 35A01.

Received: February 2016;      Revised: July 2016;      Available Online: September 2016.