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Polynomial stability of a transmission problem involving Timoshenko systems with fractional Kelvin–Voigt damping
In this work, we study the stability of a one‐dimensional Timoshenko system with localized internal fractional Kelvin–Voigt damping in a bounded domain. First, we reformulate the system into an augmented model and using a general criteria of Arendt–Batty we prove the strong stability. Next, we inves...
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Published in: | Mathematical methods in the applied sciences 2023-04, Vol.46 (6), p.7140-7179 |
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Main Authors: | , , , |
Format: | Article |
Language: | English |
Subjects: | |
Citations: | Items that this one cites Items that cite this one |
Online Access: | Get full text |
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Summary: | In this work, we study the stability of a one‐dimensional Timoshenko system with localized internal fractional Kelvin–Voigt damping in a bounded domain. First, we reformulate the system into an augmented model and using a general criteria of Arendt–Batty we prove the strong stability. Next, we investigate three cases: The first one when the damping is localized in the bending moment, the second case when the damping is localized in the shear stress, we prove that the energy of the system decays polynomially with rate
t−1$$ {t}^{-1} $$ in both cases. In the third case, the fractional Kelvin–Voigt is acting on the shear stress and the bending moment simultaneously. We show that the system is polynomially stable with energy decay rate of type
t−42−α$$ {t}^{\frac{-4}{2-\alpha }} $$, provided that the two dampings are acting in the same subinterval. The method is based on the frequency domain approach combined with multiplier technique. |
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ISSN: | 0170-4214 1099-1476 |
DOI: | 10.1002/mma.8960 |