Stability results for an elastic-viscoelastic waves interaction systems with localized Kelvin-Voigt damping and with an internal or boundary time delay

We investigate the stability of a one-dimensional wave equation with non smooth localized internal viscoelastic damping of Kelvin-Voigt type and with boundary or localized internal delay feedback. The main novelty in this paper is that the Kelvin-Voigt and the delay damping are both localized via non smooth coefficients. In the case that the Kelvin-Voigt damping is localized faraway from the tip and the wave is subjected to a locally distributed internal or boundary delay feedback, we prove that the energy of the system decays polynomially of type t^{-4}. However, an exponential decay of the energy of the system is established provided that the Kelvin-Voigt damping is localized near a part of the boundary and a time delay damping acts on the second boundary. While, when the Kelvin-Voigt and the internal delay damping are both localized via non smooth coefficients near the tip, the energy of the system decays polynomially of type t^{-4}. Frequency domain arguments combined with piecewise multiplier techniques are employed.