Uniform decay rate estimates for the wave equation in an inhomogeneous medium with simultaneous interior and boundary feedbacks

Abstract

Inspired by the highly cited work due to Lasiecka and Tataru (1993) [20], a semilinear model of the wave equation in an inhomogeneous medium with simultaneous interior and boundary feedbacks is considered. While for a homogeneous medium Ω and for certain geometries nonlinear feedback is strong enough to derive uniform decay rates of the energy associated with the model, in contrast, in an inhomogeneous medium, and for general geometry of Ω, the problem requires a new approach. In the first case, the rays of the geometric optics are straight lines and the geometry is favorable. However, in the second one, more damping is required to well-known obtain a similar result. For this purpose, arguments of microlocal analysis are used instead of the multipliers technique.