Local uniform stability for the semilinear wave equation in inhomogeneous media with locally distributed Kelvin–Voigt damping

We consider the semilinear wave equation posed in an inhomogeneous medium Ω with smooth boundary ∂Ω subject to a local viscoelastic damping distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. We show that the energy of the wave equation goes uniformly an...

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Bibliographic Details
Published inMathematische Nachrichten Vol. 291; no. 14-15; pp. 2145 - 2159
Main Authors Astudillo, M., Cavalcanti, M. M., Fukuoka, R., Gonzalez Martinez, V. H.
Format Journal Article
LanguageEnglish
Published Weinheim Wiley Subscription Services, Inc 01.10.2018
Subjects
35L05
35l20
exponential stability
Inhomogeneous media
Kelvin–Voigt damping
MSC 35B40
Smooth boundaries
Viscoelastic damping
Viscoelasticity
wave equation
Wave equations
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Summary:We consider the semilinear wave equation posed in an inhomogeneous medium Ω with smooth boundary ∂Ω subject to a local viscoelastic damping distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. We show that the energy of the wave equation goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase‐space. As far as we know, this is the first stabilization result for a semilinear wave equation with localized Kelvin–Voigt damping.
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.201700109