On the exponential time-decay for the one-dimensional wave equation with variable coefficients

TL;DR

This paper proves that the local energy of solutions to the one-dimensional wave equation with variable coefficients decays exponentially over time, providing explicit decay rates and constants using two different analytical techniques.

Contribution

It establishes exponential decay of local energy for the wave equation with variable coefficients and offers explicit decay rates and constants, improving understanding of energy dissipation in such systems.

Findings

Exponential decay of local energy proven for wave equations with variable coefficients.

Explicit decay constants provided for large-time behavior.

Two methods yield decay estimates, with one offering improved rates under certain conditions.

Abstract

We consider the initial-value problem for the one-dimensional, time-dependent wave equation with positive, Lipschitz continuous coefficients, which are constant outside a bounded region. 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 for large times. We give explicit estimates of the rate of this exponential decay by two different techniques. The first one is based on the definition of a modified, weighted local energy, with suitably constructed weights. The second one is based on the integral formulation of the problem and, under a more restrictive assumption on the variation of the coefficients, allows us to obtain improved decay rates.