A nonlocal two-phase Stefan problem. (English) Zbl 1313.35352

Chasseigne, Emmanuel; Sastre-Gómez, Silvia

Differ. Integral Equ. 26, No. 11-12, 1335-1360 (2013).

In this paper, a nonlocal version of the two-phase Stefan problem is studied. This new nonlocal mathematical model of a phase-transition problem explains the creation and evolution of mushy regions. The authors investigate sign-changing solutions of the equation, which involves a convolution with a continuous nonnegative compactly supported kernel. The basic theory of the model, including existence and uniqueness results, is developed for integrable initial data and for continuous initial data.
The asymptotic behavior of sign-changing solutions is investigated in several particular cases. The authors give a criterion which ensures that positive and negative phases will never interact. Then, the asymptotic behavior is given separately by each phase. For the case when some interactions between phases can occur, but only in the mushy region, the authors prove that the asymptotic behavior can be described by a bi-obstacle problem, which has a unique solution in a suitable class.

Reviewer: Elena Frolova (St. Petersburg)

MSC:

35R09	Integro-partial differential equations
45K05	Integro-partial differential equations
45M05	Asymptotics of solutions to integral equations

Keywords:

nonlocal equation; two-phase Stefan problem; mushy regions; asymptotic behavior