Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions

Wehbe Ali; Nasser Rayan; Noun Nahla

doi:10.3934/mcrf.2020050

Article Contents

2021, Volume 11, Issue 4: 885-904. Doi: 10.3934/mcrf.2020050 open access

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Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions

1.
Université de Bretagne Occidentale, LMBA, Brest, France, Lebanese University, Faculty of Sciences, Khawarizmi Laboratory of Mathematics and Applications-KALMA, Hadath-Beirut, Lebanon
2.
Lebanese University, Faculty of Sciences, Khawarizmi Laboratory of Mathematics and Applications-KALMA, Hadath-Beirut, Lebanon

^* Corresponding author: Ali Wehbe
^* Corresponding author: Ali Wehbe

Received: February 1, 2020

Revised: September 1, 2020

Early access: December 10, 2020

Published online: December 1, 2021

The first author is supported by the CNRS and the LAMA laboratory of Mathematics of the Université Savoie Mont Blanc.

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Abstract

We investigate a multidimensional transmission problem between viscoelastic system with localized Kelvin-Voigt damping and purely elastic system under different types of geometric conditions. The Kelvin-Voigt damping is localized via non smooth coefficient in a suitable subdomain. It was shown that the discontinuity of the material coefficient along the interface elastic/viscoelastic can't assure an exponential stability of the total system. So, it is natural to hope for a polynomial stability result under certain geometric conditions on the damping region. For this aim, using frequency domain approach combined with a new multiplier technic, we will establish a polynomial energy decay estimate of type $ t^{-1} $ for smooth initial data. This result is obtained if either one of the geometric assumptions (A1) or (A2) holds (see below). Also, we establish a general polynomial energy decay estimate on a bounded domain where the geometric conditions on the localized viscoelastic damping are violated and we apply it on a square domain where the damping is localized in a vertical strip. However, the energy of our system decays polynomially of type $ t^{-2/5} $ if the strip is localized near the boundary. Else, it's of type $ t^{-1/3} $. The main novelty in this paper is that the geometric situations covered here are richer and less restrictive than those considered in [31], [28], [19] and include in particular an example where the damping region is localized faraway from the boundary. Note that part of the results of this paper was announced in [22].

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Mathematics Subject Classification: Primary: 35B35, 93B52; Secondary: 93C20.

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Figure 1. Elastic-viscoelastic waves interaction models satisfying the assumption (A1)

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Figure 2. A model satisfying assumption (A2)

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Figure 3. A model satisfying both (A1) and (A2)

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Figure 4. A square model with local viscoelastic strip not satisfying any geometry cited before

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Figure 5. A model satisfying $ (m \cdot \nu_2)\vert _{\Gamma_2} \leq 0 $

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Figure 6. A model not satisfying $ (m \cdot \nu_2)\vert _{\Gamma_2} \leq 0 $

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