A nonlocal two-phase Stefan problem

Emmanuel Chasseigne (1, 2) , Silvia Sastre-Gomez (3)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 FDP - Fédération de recherche Denis Poisson
3 Departamento de Matematica Aplicada [Madrid]

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.

Mots clés: Stefan problem; mushy regions; nonlocal equations; degenerate parabolic equations

Domaines: Equations aux dérivées partielles [math.AP]

Emmanuel Chasseigne, Silvia Sastre-Gomez. A nonlocal two-phase Stefan problem. Differential and integral equations, 2013, 26, pp.1235-1434. ⟨hal-00841416⟩