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Uniform stabilization of some damped second order evolution equations with vanishing short memory

Published online by Cambridge University Press:  23 December 2013

Louis Tebou
Louis Tebou*
Affiliation:
Department of Mathematics and Statistics, Florida International University, Miami FL 33199, USA. teboul@fiu.edu
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Abstract

We consider a damped abstract second order evolution equation with an additional vanishing damping of Kelvin–Voigt type. Unlike the earlier work by Zuazua and Ervedoza, we do not assume the operator defining the main damping to be bounded. First, using a constructive frequency domain method coupled with a decomposition of frequencies and the introduction of a new variable, we show that if the limit system is exponentially stable, then this evolutionary system is uniformly − with respect to the calibration parameter − exponentially stable. Afterwards, we prove uniform polynomial and logarithmic decay estimates of the underlying semigroup provided such decay estimates hold for the limit system. Finally, we discuss some applications of our results; in particular, the case of boundary damping mechanisms is accounted for, which was not possible in the earlier work mentioned above.

Keywords

Second order evolution equationKelvin–Voigt dampinghyperbolic equationsstabilizationboundary dissipationlocalized dampingplate equationselasticity equationsfrequency domain methodresolvent estimates
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 1 , January 2014 , pp. 174 - 189
Copyright
© EDP Sciences, SMAI 2013

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