Asymptotic blow-up for a class of semilinear wave equations on extremal   Reissner-Nordstr\"om spacetimes
  We prove small data global existence for a class of semilinear wave equations satisfying the null condition on extremal Reissner-Nordstrom black hole backgrounds with nonlinear terms that degenerate at the event horizon. We impose no symmetry assumptions. The study of such equations is motivated by their covariance properties under the Couch-Torrence conformal isometry. We show decay, non-decay and asymptotic blow-up results analogous to those in the linear case. 
Yannis Angelopoulos and Stefanos Aretakis and Dejan Gajic
math.AP gr-qc