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

Journal Article · Sun Dec 15 00:00:00 EST 2013 · Applied Mathematics and Optimization

DOI:https://doi.org/10.1007/S00245-013-9210-8· OSTI ID:22309137

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 <##> $\}\}\mathrm{\}N}$$$ /mathcal{N} $$N. Employing standard assumptions from the theory of nonlinear unstable dynamics on the set <##> $\}\}\mathrm{\}N}$$$ /mathcal{N} $$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.

• OSTI ID: 22309137
• Journal Information: Applied Mathematics and Optimization, Vol. 68, Issue 3; Other Information: Copyright (c) 2013 Springer Science+Business Media New York; http://www.springer-ny.com; This record replaces 46015310; Country of input: International Atomic Energy Agency (IAEA); ISSN 0095-4616
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CONVERGENCE
DAMPING
EQUILIBRIUM
MATHEMATICAL SOLUTIONS
NONLINEAR PROBLEMS
PARTIAL DIFFERENTIAL EQUATIONS
TRAJECTORIES