# Stabilization of two strongly coupled hyperbolic equations in exterior domains

## TL;DR

This paper investigates the energy decay and boundedness of solutions for two coupled hyperbolic equations in exterior domains, demonstrating uniform decay and exponential energy decay under certain damping and coupling conditions.

## Contribution

It establishes conditions under which the total energy decays uniformly and the solutions remain bounded, including exponential decay for coupled Klein-Gordon equations with equal speeds.

## Key findings

- Total energy decays uniformly when damping includes the coupling set.
- The $L^2$-norm of solutions remains bounded under specified conditions.
- Exponential energy decay occurs for coupled Klein-Gordon equations with equal speeds.

## Abstract

In this paper we study the behavior of the total energy and the $L^2$-norm of solutions of two coupled hyperbolic equations by velocities in exterior domains. Only one of the two equations is directly damped by a localized damping term. We show that, when the damping set contains the coupling one and the coupling term is effective at infinity and on captive region, then the total energy decays uniformly and the $L^2$-norm of smooth solutions is bounded. In the case of two Klein-Gordon equations with equal speeds we deduce an exponential decay of the energy.