  It is well-known that small, regular, spherically symmetric characteristic
initial data to the Einstein-scalar-field system which are decaying towards
(future null) infinity give rise to solutions which are foward-in-time global
(in the sense of future causal geodesic completeness). We construct a class of
spherically symmetric solutions which are global but the initial norms are
consistent with initial data not decaying towards infinity. This gives the
following consequences: (1) We prove that there exist foward-in-time global
solutions with arbitrarily large (and in fact infinite) initial bounded
variation (BV) norms and initial Bondi masses. (2) While general solutions with
non-decaying data do not approach Minkowski spacetime, we show using the
results of Luk--Oh that if a sufficiently strong asymptotic flatness condition
is imposed on the initial data, then the solutions we construct (with large BV
norms) approach Minkowski spacetime with a sharp inverse polynomial rate. (3)
Our construction can be easily extended so that data are posed at past null
infinity and we obtain solutions with large BV norms which are causally
geodesically complete both to the past and to the future. Finally, we discuss
applications of our method to construct global solutions for other nonlinear
wave equations with infinite critical norms.
