On the limiting amplitude principle for the wave equation with variable coefficients

TL;DR

This paper establishes the validity and quantifies the convergence of the limiting amplitude principle for the wave equation with variable coefficients in multiple dimensions, extending previous results and providing numerical illustrations.

Contribution

It proves the LAP for wave equations with nonconstant coefficients in 2D and 3D, extending to 1D, with quantitative estimates and numerical validation.

Findings

LAP holds for wave equations with variable coefficients in 2D and 3D.

Quantitative estimates for convergence of solutions are provided.

Numerical examples confirm theoretical results in symmetric problems.

Abstract

In this paper, we prove new results on the validity of the limiting amplitude principle (LAP) for the wave equation with nonconstant coefficients, not necessarily in divergence form. Under suitable assumptions on the coefficients and on the source term, we establish the LAP for space dimensions 2 and 3. This result is extended to one space dimension with an appropriate modification. We also quantify the LAP and thus provide estimates for the convergence of the time-domain solution to the frequency-domain solution. Our proofs are based on time-decay results of solutions of some auxiliary problems. The obtained results are illustrated numerically on radially symmetric problems in dimensions 1, 2 and 3.