Electron. J. Differential Equations,
Vol. 2017 (2017), No. 83, pp. 118.
Stabilization of the wave equation with localized compensating
frictional and KelvinVoigt dissipating mechanisms
Marcelo Cavalcanti, Valeria Domingos Cavalcanti, Louis Tebou
Abstract:
We consider the wave equation with two types of locally distributed damping
mechanisms: a frictional damping and a KelvinVoigt type damping.
The location of each damping is such that none of them alone is able
to exponentially stabilize the system; the main obstacle being that there
is a quite big undamped region. Using a combination of the multiplier
techniques and the frequency domain method, we show that a convenient interaction
of the two damping mechanisms is powerful enough for the exponential stability
of the dynamical system, provided that the coefficient of the KelvinVoigt
damping is smooth enough and satisfies a structural condition. When the
latter coefficient is only bounded measurable, exponential stability may still
hold provided there is no undamped region, else only polynomial stability is
established. The main features of this contribution are:
(i) the use of the KelvinVoigt or short memory damping as opposed to the usual
long memory type damping; this makes the problem more difficult to solve due
to the somewhat singular nature of the KelvinVoigt damping,
(ii) allowing the presence of an undamped region unlike all earlier works where
a combination of frictional and viscoelastic damping is used.
Submitted April 26, 2016. Published March 24, 2017.
Math Subject Classifications: 93D15, 35L05.
Key Words: Stabilization; wave equation; frictional damping;
KelvinVoigt damping; viscoelastic material; localized damping
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Marcelo Cavalcanti
Department of Mathematics and Statistics
Maringa State University, 87020900
Maringa PR, Brazil
email: mmcavalcanti@uem.br


Valéeria Domingos Cavalcanti
Department of Mathematics and Statistics
Maringa State University, 87020900
Maringa PR, Brazil
email: vndcavalcanti@uem.br


Louis Tebou
Department of Mathematics and Statistics
Florida International University
Modesto Maidique Campus
Miami, Florida 33199, USA
email: teboul@fiu.edu

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