  Taking into account the recent works \cite{RoTaVe:2020} and \cite{Rys:2020},
we consider a phase-change problem for a one dimensional material with a
non-local flux, expressed in terms of the Caputo derivative, which derives in a
space-fractional Stefan problem. We prove existence of a unique solution to a
phase-change problem with the fractional Neumann boundary condition at the
fixed face $x=0$, where the domain, at the initial time, consists of liquid and
solid. Then we use this result to prove the existence of a limit solution to an
analogous problem with solid initial domain, when it is not possible to
transform the domain into a cylinder.
