# Energy decay and diffusion phenomenon for the asymptotically periodic damped wave equation

## TL;DR

This paper establishes local and global energy decay for the asymptotically periodic damped wave equation, focusing on low-frequency behavior and showing it resembles a heat equation influenced by the metric's H-limit and absorption mean.

## Contribution

It provides new insights into low-frequency energy decay and the heat-like behavior of solutions in asymptotically periodic damped wave equations.

## Key findings

- Energy decay is proven both locally and globally.
- Low frequencies behave like solutions to a heat equation.
- Decay rates depend on the H-limit of the metric and mean absorption.

## Abstract

We prove local and global energy decay for the asymptotically periodic damped wave equation on the Euclidean space. Since the behavior of high frequencies is already mostly understood, this paper is mainly about the contribution of low frequencies. We show in particular that the damped wave behaves like a solution of a heat equation which depends on the H-limit of the metric and the mean value of the absorption index.