  We prove the existence and uniqueness of solutions to a one-dimensional
Stefan Problem for reflected SPDEs which are driven by space-time white noise.
The solutions are shown to exist until almost surely positive blow-up times.
Such equations can model the evolution of phases driven by competition at an
interface, with the dynamics of the shared boundary depending on the
derivatives of two competing profiles at this point. The novel features here
are the presence of space-time white noise; the reflection measures, which
maintain positivity for the competing profiles; and a sufficient condition to
make sense of the Stefan condition at the boundary. We illustrate the behaviour
of the solution numerically to show that this sufficient condition is close to
necessary.
