Local Energy Decay and Diffusive Phenomenon in a Dissipative Wave Guide

TL;DR

This paper proves local energy decay for a dissipative wave guide, showing that over time the wave behaves like a heat equation solution, using resolvent estimates despite spectral analysis challenges.

Contribution

It establishes local energy decay in dissipative wave guides and introduces a novel spectral analysis approach without relying on Riesz basis properties.

Findings

Wave energy decays locally over time.

Dissipated waves resemble heat equation solutions asymptotically.

Spectral analysis overcomes eigenvector basis limitations.

Abstract

We prove the local energy decay for the wave equation in a wave guide with dissipation at the boundary. It appears that for large times the dissipated wave behaves like a solution of a heat equation in the unbounded directions. The proof is based on resolvent estimates. Since the eigenvectors for the transverse operator do not form a Riesz basis, the spectral analysis does not trivially reduce to separate analyses on compact and Euclidean domains.