Exponential time-decay for a one dimensional wave equation with coefficients of bounded variation

TL;DR

This paper proves that solutions to a one-dimensional wave equation with BV coefficients exhibit exponential decay of local energy over time, using resolvent estimates for a related Helmholtz operator.

Contribution

It introduces a novel exponential decay result for wave equations with BV coefficients and provides explicit decay constants, advancing understanding of wave behavior in variable media.

Findings

Local energy decays exponentially fast in time

Explicit decay constant provided

High frequency resolvent estimates established

Abstract

We consider the initial-value problem for a one-dimensional wave equation with coefficients that are positive, constant outside of an interval, and have bounded variation (BV). Under the assumption of compact support of the initial data, we prove that the local energy decays exponentially fast in time, and provide the explicit constant to which the solution converges. The key ingredient of the proof is a high frequency resolvent estimate for an associated Helmholtz operator with a BV potential.