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


GÜVEN GEREDELİ P. , Webster J. T.

APPLIED MATHEMATICS AND OPTIMIZATION, vol.68, no.3, pp.361-390, 2013 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 68 Issue: 3
  • Publication Date: 2013
  • Doi Number: 10.1007/s00245-013-9210-8
  • Title of Journal : APPLIED MATHEMATICS AND OPTIMIZATION
  • Page Numbers: pp.361-390

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 . Employing standard assumptions from the theory of nonlinear unstable dynamics on the set , 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.