Stability results for an elastic–viscoelastic wave equation with localized Kelvin–Voigt damping and with an internal or boundary time delay

Abstract

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. Under sufficient assumptions, in the case that the Kelvin–Voigt damping is localized faraway from the tip and the wave is subjected to a 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 boundary, under sufficient assumptions, using frequency domain arguments combined with piecewise multiplier techniques, we prove that the energy of the system decays polynomially of type t − 4 . Otherwise, if the above assumptions are not true, we establish instability results.