Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping

Louis Tebou

doi:10.3934/dcds.2016110

Article Contents

2016, Volume 36, Issue 12: 7117-7136. Doi: 10.3934/dcds.2016110

This issue Previous Article Infinitely many solutions for a nonlinear Schrödinger equation with non-symmetric electromagnetic fields Next Article Existence and concentration of solutions for a Kirchhoff type problem with potentials

Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping

Louis Tebou^1,

1.
Department of Mathematics & Statistics, Florida International University, Miami, FL 33199

Received: October 31, 2015

Revised: July 31, 2016

Published: May 2017

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Abstract

We consider the dynamic elasticity equations with a locally distributed damping of Kelvin-Voigt type in a bounded domain. The damping is localized in a suitable open subset, of the domain under consideration, which satisfies the piecewise multipliers condition of Liu. Using multiplier techniques combined with the frequency domain method, we show that: i) the energy of this system decays polynomially when the damping coefficient is only bounded measurable, ii) the energy of this system decays exponentially when the damping coefficient as well as its gradient are bounded measurable, and the damping coefficient further satisfies a structural condition. These results generalize and improve, at the same time, on an earlier result of Liu and Rao involving the wave equation with Kelvin-Voigt damping; those authors proved the exponential decay of the energy provided that the damping region is a neighborhood of the whole boundary, and further restrictions are imposed on the damping coefficient.

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Mathematics Subject Classification: Primary: 93D15; Secondary: 74J05, 35L05, 74H40.

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