We construct solutions with prescribed radiation fields for wave equations with
polynomially decaying sources close to the light cone. In this setting, which is
motivated by semilinear wave equations satisfying the weak null condition, solutions
to the forward problem have a logarithmic leading order term on the light cone and
nontrivial homogeneous asymptotics in the interior of the light cone. The
backward scattering solutions we construct are given to second order by
explicit asymptotic solutions in the wave zone, and in the interior of the light
cone which satisfy novel matching conditions. In the process we find novel
compatibility conditions for the scattering data at null infinity. We also relate
the asymptotics of the radiation field towards space-like infinity to explicit
homogeneous solutions in the exterior of the light cone. This is the setting
of slowly polynomially decaying data corresponding to mass, charge and
angular momentum in applications. We show that homogeneous data of degree
and
for
the wave equation results in the same logarithmic terms on the light cone and
homogeneous asymptotics in the interior as for the equations with sources
close to the light cone. The proof requires a delicate analysis of the forward
solution close to the light cone and uses the invertibility of the Funk transform.
