  We study a nonlocal version of the two-phase Stefan problem, which models a
phase transition problem between two distinct phases evolving to distinct heat
equations. Mathematically speaking, this consists in deriving a theory for
sign-changing solutions of the equation, ut = J * v - v, v = {\Gamma}(u), where
the monotone graph is given by {\Gamma}(s) = sign(s)(|s|-1)+ . We give general
results of existence, uniqueness and comparison, in the spirit of [2]. Then we
focus on the study of the asymptotic behaviour for sign-changing solutions,
which present challenging difficulties due to the non-monotone evolution of
each phase.

Analysis of PDEs