Blow-up for semilinear wave equations on Kerr black hole backgrounds

TL;DR

This paper proves finite-time blow-up for solutions to semilinear wave equations on Kerr black hole backgrounds, showing black holes do not prevent singularity formation in certain regions, and extends results to general asymptotically flat spacetimes.

Contribution

It establishes blow-up results for semilinear wave equations on Kerr and Schwarzschild backgrounds, and demonstrates the approach's applicability to a broad class of asymptotically flat spacetimes.

Findings

Solutions blow up along outgoing null cones for small initial data.

Black hole backgrounds do not prevent singularity formation away from horizons.

Method extends to various asymptotically flat space-times.

Abstract

We examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the blow up part of the John theorem, with $F_p(u)=|u|^{p}$, on the Schwarzschild and Kerr black hole backgrounds. Concerning the case of Schwarzschild, we construct a class of small data, so that the solution blows up along the outgoing null cone, which applies for both $F_p(u)=|u|^{p}$ and the focusing nonlinearity $F_p(u)=|u|^{p-1}u$. The proof suggests that the black hole does not have any essential influence on the formation of singularity, in the region away from the Cauchy horizon $r=r_-$ or the singularity $r=0$. Our approach is also robust enough to be adapted for general asymptotically flat space-time manifolds, possibly exterior to a compact domain, with spatial dimension $n\ge 2$. Typical examples include exterior domains, asymptotically Euclidean spaces, Reissner-N\"ordstr\"om space-times, and Kerr-Newman space-times.