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We study spherically symmetric solutions of semilinear wave equations in thecase where the nonlinearity satisfies the null condition on extremalReissner--Nordstrom black hole spacetimes. We show that solutions which arisefrom sufficiently small compactly supported smooth data prescribed on a Cauchyhypersfurace \widetilde{{\Sigma}}_0 crossing the future event horizon\mathcal{H}^{+} are globally well-posed in the domain of outer communicationsup to and including \mathcal{H}^{+}. Our method allows us to close allbootstrap estimates under very weak decay results (compatible with those knownfor the linear case). Moreover we establish a certain number of non-decay andblow-up results along the horizon \mathcal{H}^{+} which generalize knowninstability results for the linear case. Our results apply to sphericallysymmetric wave maps for a wide class of target spaces.

Nonlinear Wave Equations With Null Condition On Extremal Reissner-Nordstr\"om Spacetimes I: Spherical Symmetry