Generic blow-up results for the wave equation in the interior of a Schwarzschild black hole

TL;DR

This paper analyzes the asymptotic behavior of solutions to the wave equation inside Schwarzschild black holes, revealing detailed blow-up characteristics and providing a comprehensive expansion near the singularity without symmetry assumptions.

Contribution

It provides the first full asymptotic expansion for solutions near the black hole singularity and characterizes initial data leading to logarithmic blow-up, using energy estimates without symmetry constraints.

Findings

Solutions exhibit a principal logarithmic term near the singularity.

A subset of initial data causes solutions to blow up logarithmically.

The method involves weighted energy estimates and analysis of spherically symmetric components.

Abstract

We study the behaviour of smooth solutions to the wave equation, $\square_g\psi=0$, in the interior of a fixed Schwarzschild black hole. In particular, we obtain a full asymptotic expansion for all solutions towards $r=0$ and show that it is characterised by its first two leading terms, the principal logarithmic term and a bounded second order term. Moreover, we characterise an open set of initial data for which the corresponding solutions blow up logarithmically on the entirety of the singular hypersurface $\{r=0\}$. Our method is based on deriving weighted energy estimates in physical space and requires no symmetries of solutions. However, a key ingredient in our argument uses a precise analysis of the spherically symmetric part of the solution and a monotonicity property of spherically symmetric solutions in the interior.