Energy Decay in a Wave Guide with Dissipation at Infinity

TL;DR

This paper proves local and global energy decay for the wave equation in a wave guide with damping at infinity, revealing diffusive phenomena and addressing cases where the geometric control condition is not met.

Contribution

It establishes new energy decay results for wave equations with damping at infinity, including cases without the geometric control condition, in both wave guides and Euclidean spaces.

Findings

Energy decay is proven for wave guides with damping at infinity.

Diffusive phenomena are observed for low frequencies.

Results include cases where the geometric control condition fails.

Abstract

We prove local and global energy decay for the wave equation in a wave guide with damping at infinity. More precisely, the absorption index is assumed to converge slowly to a positive constant, and we obtain the diffusive phenomenon typical for the contribution of low frequencies when the damping is effective at infinity. On the other hand, the usual Geometric Control Condition is not necessarily satisfied so we may have a loss of regularity for the contribution of high frequencies. Since our results are new even in the Euclidean space, we also state a similar result in this case.