Uniform Stabilization for the Semi-linear Wave Equation with Nonlinear Kelvin–Voigt Damping

This paper is concerned with the decay estimate of solutions to the semilinear wave equation subject to two localized dampings in a bounded domain. The first one is of the nonlinear Kelvin–Voigt type which is distributed around a neighborhood of the boundary and the second is a frictional damping depending in the first one. We show uniform decay rate results of the corresponding energy for all initial data taken in bounded sets of finite energy phase-space. The proof is based on obtaining an observability inequality which combines unique continuation properties and the tools of the Microlocal Analysis Theory