Uniform stabilization for the Klein-Gordon system in a inhomogeneous medium with locally distributed damping

TL;DR

This paper proves that the energy of the Klein-Gordon system in an inhomogeneous medium with localized damping decays uniformly and exponentially, using advanced microlocal analysis and Carleman estimates.

Contribution

It introduces a novel combination of microlocal analysis and Carleman estimates to establish exponential stabilization for Klein-Gordon systems with localized damping.

Findings

Energy decays exponentially to zero

Uniform stabilization holds for all finite energy initial data

Refined microlocal analysis and Carleman estimates are effective

Abstract

We consider the Klein-Gordon system posed in an inhomogeneous medium with smooth boundary subject to a local viscoelastic damping distributed around a neighborhoodof the boundary according to the Geometric Control Condition. We show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space. For this purpose, refined microlocal analysis arguments are considered by exploiting ideas due to Burq and Gerard . By using sharp Carleman estimates we prove a unique continuation property for coupled systems.