Global Spherically Symmetric Solutions of Non-linear Wave Equations with Null Condition on Extremal Reissner-Nordstrom Spacetimes

We study spherically symmetric solutions of semilinear wave equations in the case where the non-linearity satisfies the null condition on extremal Reissner-Nordstrom black hole spacetimes. We show that solutions that arise from sufficiently small compactly supported smooth data prescribed on a Cauchy hypersurface (Sigma) over tilde (0) crossing the future event horizon H+ are globally well posed in the domain of outer communications up to and including H+. Our method allows us to close all bootstrap estimates under very weak decay results (compatible with those known for the linear case). Moreover, we establish a certain number of non-decay and blow-up results along the horizon H+ which generalize known instability results for the linear case. Our results apply to spherically symmetric wave maps for a wide class of target spaces.