Stabilization for the Wave Equation with Singular Kelvin–Voigt Damping

We consider the wave equation with Kelvin–Voigt damping in a bounded domain. The exponential stability result proposed by Liu and Rao (Z Angew Math Phys (ZAMP) 57:419–432, 2006) or Tebou (C R Acad Sci Paris Ser I 350: 603–608, 2012) for that system assumes that the damping is localized in a neighborhood of the whole or a part of the boundary under some consideration. In this paper we propose to deal with this geometrical condition by considering a singular Kelvin–Voigt damping which is localized far away from the boundary. In this particular case Liu and Liu (SIAM J Control Optim 36:1086–1098, 1998) proved the lack of the uniform decay of the energy. However, we show that the energy of the wave equation decreases logarithmically to zero as time goes to infinity. Our method is based on the frequency domain method. The main feature of our contribution is to write the resolvent problem as a transmission system to which we apply a specific Carleman estimate.