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

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.