# Behaviors of the energy of solutions of two coupled wave equations with nonlinear damping on a compact manifold with boundary

## TL;DR

This paper investigates how the energy of solutions to coupled wave equations with nonlinear damping on a compact manifold decays over time, revealing that decay rates are governed by a specific differential equation.

## Contribution

It establishes decay rate estimates for coupled wave equations with nonlinear damping, under geometric conditions, extending understanding of indirect damping effects on energy decay.

## Key findings

- Energy decay rates are characterized by a first order differential equation.
- Decay behavior depends on geometric conditions of coupling and damping regions.
- Results apply to smooth solutions on compact manifolds with boundary.

## Abstract

In this paper we study the behaviors of the the energy of solutions of coupled wave equations on a compact manifold with boundary in the case of indirect nonlinear damping . Only one of the two equations is directly damped by a localized nonlinear damping term. Under geometric conditions on both the coupling and the damping regions we prove that the rate of decay of the energy of smooth solutions of the system is determined from a first order differential equation .