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| An Extension of the rp Method for WaveEquations with Scale-Critical Potentials andFirst-Order Terms |
| Maxime Van de Moortel |
| (Department of Mathematics, Rutgers University, Hill Center, New Brunswick,
NJ 08854, USA) |
| DOI： |
| Abstract: |
| The r
p method, first introduced in [9], 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 Gr¨onwall argument, which allows a larger range of p than the standard
method. A spherically symmetric version of our strategy first appeared in [22]
in the context of a weakly charged scalar field on a black hole whose equations
also involve a scale-critical potential. |
| Key words: Scale-critical potential, r
p method |