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

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An Erratum to this article was published on 11 October 2014

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.

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Acknowledgements

Justin Webster was partially supported by NASA’s Virginia Space Grant Consortium Graduate Research Fellowship NNX10AT94H, 2011–2013.

The authors would like to thank the anonymous referee for the valuable feedback which improved the clarity and content of the treatment.

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Authors and Affiliations

  1. Hacettepe University, Ankara, Turkey

    Pelin G. Geredeli

  2. Oregon State University, Corvallis, OR, USA

    Justin T. Webster

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  1. Pelin G. Geredeli
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  2. Justin T. Webster
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Correspondence to Justin T. Webster.

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Geredeli, P.G., Webster, J.T. Decay Rates to Equilibrium for Nonlinear Plate Equations with Degenerate, Geometrically-Constrained Damping. Appl Math Optim 68, 361–390 (2013). https://doi.org/10.1007/s00245-013-9210-8

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