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

  • * Corresponding author: Ali Wehbe

    * Corresponding author: Ali Wehbe 

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

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  • 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].

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

    Citation:

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

    Figure 2.  A model satisfying assumption (A2)

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

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

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

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

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