  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.
