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

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