A nonlocal two phase Stefan problem

Abstract

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, $u_t=J\ast v -v $, $v=\Gamma(u)$, where the monotone graph is given by $\Gamma(s)=\rm{sign}(s)(|s|-1)_+$. We give general results of existence, uniqueness and comparison, in the spirit of [2]. Then we focus on the study of the asymptotic behavior for sign-changing solutions, which present challenging difficulties due to the nonmonotone evolution of each phase.