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

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.

KEYWORDS:

• wave equation
• variable coefficients
• limiting amplitude principle
• long-time asymptotics

MSC SUBJECT CLASSIFICATION (2010):

• 35L05
• 35L10
• 35B10
• 35B40

Acknowledgments

We would like to express our gratitude to the anonymous reviewer whose suggestions have helped improving the manuscript.

Additional information

Funding

This work was supported by the Austrian Science Fund (FWF) projects 10.55776/F65 (AA, SG, IP), 10.55776/P33477 (IP), and 10.55776/I3538 (AA, DP).