Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay

Mohammad Akil; Haidar Badawi; Ali Wehbe

doi:10.3934/cpaa.2021092

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2021, Volume 20, Issue 9: 2991-3028. Doi: 10.3934/cpaa.2021092

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Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay

1.
Université Savoie Mont Blanc, LAMA, Chambéry, France
2.
Université Polytechnique Hauts-de-France, LAMAV, Valenciennes, France
3.
Lebanese University, Faculty of sciences, Khawarizmi Laboratory of Mathematics and Applications-KALMA, Hadath-Beirut, Lebanon

^* Corresponding author
^* Corresponding author

Received: January 31, 2021

Revised: March 31, 2021

Early access: June 2021

Published: September 2021

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Abstract

The purpose of this paper is to investigate the stabilization of a locally coupled wave equations with non smooth localized viscoelastic damping of Kelvin-Voigt type and localized time delay. Using a general criteria of Arendt-Batty, we show the strong stability of our system in the absence of the compactness of the resolvent. Finally, using frequency domain approach combined with the multiplier method, we prove a polynomial energy decay rate of order $ t^{-1} $.

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Mathematics Subject Classification: Primary: 93C43, 93D15, 93D20; Secondary: 35L05, 93C80.

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Figure 1. Local Kelvin-Voigt damping and local time delay feedback

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Figure 2. Geometric description of the functions $ \chi_1 $ and $ \chi_2 $

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Figure 3. Geometric description of the functions $ \theta_1 $, $ \theta_2 $ and $ \theta_3 $

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Figure 4. Geometric description of the functions $ \theta_4 $ and $ \theta_5 $

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