  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.
