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

Anton Arnold (1) , Sjoerd Geevers (2) , Ilaria Perugia (2) , Dmitry Ponomarev (3, 4, 1)

1 TU Wien - Vienna University of Technology = Technische Universität Wien
2 University of Vienna [Vienna]
3 FACTAS - Analyse fonctionnelle pour la conception et l'analyse de systèmes
4 PDMI RAS - St. Petersburg Department of V.A. Steklov Mathematical Institute

In this paper, we prove new results on the validity of the limiting ampitude 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.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse classique [math.CA]; Formation de Structures et Solitons [nlin.PS]; Systèmes Solubles et Intégrables [nlin.SI]; Acoustique [physics.class-ph]; Optique [physics.optics]

Anton Arnold, Sjoerd Geevers, Ilaria Perugia, Dmitry Ponomarev. On the limiting amplitude principle for the wave equation with variable coefficients. 2023. ⟨hal-03900877v2⟩