Semilinear Wave Equations on Extremal Reissner–Nordström Black Holes Revisited

Abstract

We revisit global existence and decay for small-data solutions of semilinear wave equations on extremal Reissner–Nordström black hole backgrounds satisfying the classical null condition, a problem which was previously addressed by the first author in joint work with Angelopoulos et al. (Ann PDE 6(2):12, 2020). In this paper, we develop a new approach based on propagating a significantly weaker set of estimates, which allows for a simpler and more streamlined proof. Our proof does not require tracking sharp estimates for the solution in the near-horizon region, which means that it is compatible with, but does not imply, the non-decay and growth hierarchy of derivatives of the solution along the event horizon expected from the Aretakis instability. In particular, this approach is in principle compatible with other settings where stronger horizon instabilities are expected, such as nonlinear charged scalar fields on extremal Reissner–Nordström, or nonlinear waves on extremal Kerr. We also sketch how our proof applies to semilinear problems on spacetimes settling down to extremal Reissner–Nordström, such as those constructed in our joint work with Angelopoulos et al. (Nonlinear stability of extremal Reissner–Nordström black holes in spherical symmetry, 2024. arXiv:2410.16234).