An extension of the r^p method for wave equations with scale-critical potentials and first-order terms

2025/05/19 by Maxime Van de Moortel, Van de Moortel, Maxime · 1 citation
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2505.13711

openalex publication_date 2025/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The r^p method, first introduced in [DR10], has become a robust strategy to prove decay for wave equations in the context of black holes and beyond. In this note, we propose an extension of this method, which is particularly suitable for proving decay for a general class of wave equations featuring a scale-critical time-dependent potential and/or first-order terms of small amplitude. Our approach consists of absorbing error terms in the r^p-weighted energy using a novel Grönwall argument, which allows a larger range of p than the standard method. A spherically symmetric version of our strategy first appeared in [VdM22] in the context of a weakly charged scalar field on a black hole whose equations also involve a scale-critical potential.

Citations

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Cited by

• Boundedness and decay of waves on spatially flat decelerated FLRW spacetimes

Related

• A new physical-space approach to decay for the wave equation with applications to black hole spacetimes
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• Universality of blow up profile for small blow up solutions to the\n energy critical wave map equation
• Blowup stability at optimal regularity for the critical wave equation