A nonlocal two-phase Stefan problem

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 = Γ(u), where the monotone graph is given by Γ(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