Decay Rates to Equilibrium for Nonlinear Plate Equations with Degenerate, Geometrically-Constrained Damping
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Geredeli, Pelin G., and Justin T. Webster. "Decay Rates to Equilibrium for Nonlinear Plate Equations with Degenerate, Geometrically-Constrained Damping." Applied Mathematics & Optimization 68, no. 3 (December 1, 2013): 361–90. https://doi.org/10.1007/s00245-013-9210-8.
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This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature's AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s00245-013-9210-8
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. 254:1193–29, 2013), we have that any trajectory converges to the set of stationary points N. Employing standard assumptions from the theory of nonlinear unstable dynamics on the set 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.