Nonlinear Waves on Extremal Black Hole Spacetimes

In this thesis I initiate the study of the global behaviour of solutions of nonlinear wave equations where the nonlinearity satisfies the null condition on extremal Reissner-Nordstrom black hole spacetimes.Under the assumption of spherical symmetry I show that solutions which arise from sufficiently small compactly supported smooth data prescribed on a Cauchy hypersurface \widetilde{\Sigma}_0 crossing the future event horizon \mathcal{H}^{+} are globally well-posed in the domain of outer communications up to and including \mathcal{H}^{+} for derivative nonlinearities of quadratic nature satisfying the null condition.Without the assumption of spherical symmetry I prove the same result for nonlinearities of quartic nature that satisfy the null condition as well.In both situations a certain number of non-decay and blow-up results along the horizon \mathcal{H}^{+} are established which generalize known instability results for the linear case. These asymptotic instabilities constitute a completely new feature that is shown for the first time in a nonlinear setting.These questions arise as model problems for the more general problem of stability or instability of extremal black hole spacetimes in the context of the Einstein equations. The results provided in this thesis can be considered as a first step towards the understanding of the fully nonlinear problem.Ph.D