On the interaction of metric trapping and a boundary

TL;DR

This paper investigates how metric trapping interacts with boundaries in a symmetric warped product manifold, revealing a bifurcation that affects wave energy decay and trapping stability.

Contribution

It introduces a specific example demonstrating the bifurcation caused by boundary movement through trapped sets, linking trapping stability to boundary position.

Findings

Transition from nontrapping to trapped rays as boundary passes through trapped set

Loss of decay estimates in stably trapped scenarios

Identification of bifurcation point affecting wave energy decay

Abstract

By considering a two ended warped product manifold, we demonstrate a bifurcation that can occur when metric trapping interacts with a boundary. In this highly symmetric example, as the boundary passes through the trapped set, one goes from a nontrapping scenario where lossless local energy estimates are available for the wave equation to the case of stably trapped rays where all but a logarithmic amount of decay is lost.