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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ t^{-1} $\end{document}</tex-math></inline-formula> for smooth initial data. This result is obtained if either one of the geometric assumptions (A1) or (A2) holds (see below). Also, we establish a general polynomial energy decay estimate on a bounded domain where the geometric conditions on the localized viscoelastic damping are violated and we apply it on a square domain where the damping is localized in a vertical strip. However, the energy of our system decays polynomially of type <inline-formula><tex-math id="M2">\begin{document}$ t^{-2/5} $\end{document}</tex-math></inline-formula> if the strip is localized near the boundary. Else, it's of type <inline-formula><tex-math id="M3">\begin{document}$ t^{-1/3} $\end{document}</tex-math></inline-formula>. The main novelty in this paper is that the geometric situations covered here are richer and less restrictive than those considered in [<xref ref-type="bibr" rid="b31">31</xref>], [<xref ref-type="bibr" rid="b28">28</xref>], [<xref ref-type="bibr" rid="b19">19</xref>] 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 [<xref ref-type="bibr" rid="b22">22</xref>].</p>