  We study spherically symmetric solutions of semilinear wave equations in the
case where the nonlinearity satisfies the null condition on extremal
Reissner--Nordstrom black hole spacetimes. We show that solutions which arise
from sufficiently small compactly supported smooth data prescribed on a Cauchy
hypersfurace \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}^{+}. 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 \mathcal{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.
