# Decay Rates to Equilibrium for Nonlinear Plate Equations with Degenerate, Geometrically-Constrained Damping

Decay Rates to Equilibrium for Nonlinear Plate Equations with Degenerate,... We analyze the convergence to equilibrium of solutions to the nonlinear Berger plate evolution equation in the presence of localized interior damping (also referred to as geometrically constrained damping ). Utilizing the results in (Geredeli et al. in J. Differ. Equ. 254:1193–1229, 2013 ), we have that any trajectory converges to the set of stationary points $\mathcal{N}$ . Employing standard assumptions from the theory of nonlinear unstable dynamics on the set $\mathcal{N}$ , we obtain the rate of convergence to an equilibrium. The critical issue in the proof of convergence to equilibria is a unique continuation property (which we prove for the Berger evolution) that provides a gradient structure for the dynamics. We also consider the more involved von Karman evolution, and show that the same results hold assuming a unique continuation property for solutions, which is presently a challenging open problem. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Decay Rates to Equilibrium for Nonlinear Plate Equations with Degenerate, Geometrically-Constrained Damping

, Volume 68 (3) – Dec 1, 2013
30 pages

/lp/springer_journal/decay-rates-to-equilibrium-for-nonlinear-plate-equations-with-0bsVp0rxHQ
Publisher
Springer US
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-013-9210-8
Publisher site
See Article on Publisher Site

### Abstract

We analyze the convergence to equilibrium of solutions to the nonlinear Berger plate evolution equation in the presence of localized interior damping (also referred to as geometrically constrained damping ). Utilizing the results in (Geredeli et al. in J. Differ. Equ. 254:1193–1229, 2013 ), we have that any trajectory converges to the set of stationary points $\mathcal{N}$ . Employing standard assumptions from the theory of nonlinear unstable dynamics on the set $\mathcal{N}$ , we obtain the rate of convergence to an equilibrium. The critical issue in the proof of convergence to equilibria is a unique continuation property (which we prove for the Berger evolution) that provides a gradient structure for the dynamics. We also consider the more involved von Karman evolution, and show that the same results hold assuming a unique continuation property for solutions, which is presently a challenging open problem.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Dec 1, 2013

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