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 com-pact domain, with spatial dimension n ≥ 2. Typical examples include exterior domains, asymptotically Euclidean spaces, Reissner-N¨ordstr¨om space-times, and Kerr-Newman space-times.