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Energy decay of some boundary coupled systems involving wave\ Euler-Bernoulli beam with one locally singular fractional Kelvin-Voigt damping

  • * Corresponding author: Mohammad Akil

    * Corresponding author: Mohammad Akil 
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  • In this paper, we investigate the energy decay of hyperbolic systems of wave-wave, wave-Euler-Bernoulli beam and beam-beam types. The two equations are coupled through boundary connection with only one localized non-smooth fractional Kelvin-Voigt damping. First, we reformulate each system into an augmented model and using a general criteria of Arendt-Batty, we prove that our models are strongly stable. Next, by using frequency domain approach, combined with multiplier technique and some interpolation inequalities, we establish different types of polynomial energy decay rate which depends on the order of the fractional derivative and the type of the damped equation in the system.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  (EBB)-W$ _{FKV} $ Model

    Figure 2.  W-(EBB)$ _{FKV} $ Model

    Figure 3.  W-W$ _{FKV} $ Model

    Figure 4.  (EBB) $ _{FKV} $ Model

    Figure 5.  (EBB)-(EBB)$ _{FKV} $

    Table 1.  Decay Results

    Model Decay Rate $ \alpha\to 1 $
    (EBB)-W$ _{FKV} $ $ t^{\frac{-4}{2-\alpha}} $ $ t^{-4} $
    W-W$ _{FKV} $ $ t^{\frac{-4}{2-\alpha}} $ $ t^{-4} $
    W-(EBB)$ _{FKV} $ $ t^{\frac{-2}{3-\alpha}} $ $ t^{-1} $
    (EBB)$ _{FKV} $ $ t^{\frac{-2}{1-\alpha}} $ Exponential
    (EBB)-(EBB)$ _{FKV} $ $ t^{\frac{-2}{3-\alpha}} $ $ t^{-1} $
     | Show Table
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