A Stabilization for a Coupled Wave System with Nonlinear and Arbitrary Damping

Aims/ Objectives: In this paper, based on motivations coming from various physical applications, we consider a coupled system of the wave in a one-dimensional bounded domain with nonlinear localized damping acting in their equations. We also discuss the well-posedness and smoothness of solutions using the nonlinear semigroup theory. Then, we give the asymptotic stability and rates decay to the coupled system, based on solution of an ordinary differential equation, since the feedback functions and the localized functions satisfy some properties widely treated in obtaining uniform decay rates for solutions of semilinear wave equation. Furthermore, the result requires the obtaining of the internal observability inequality for the conservative system.