On the Stefan Problems for the System of Equations Arising in the Modelling of Liquid-Phase Epitaxy Processes

Abstract

One-dimensional one-phase supercooled Stefan problems for the system of two diffusion equations connected with each other under the free boundary conditions are considered. In the problems of the I-st type called “direct” the values of functions on the free boundary are known while in the problems of the II-nd type called “inverse” these values should be determined with the use of an additional condition. The problems of these two types are considered in the cases of a restricted domain occupied by phase (0<x<S(t)) and a semi-infinite one (S(t)<x<∞). For “direct” problems the possibilities of time-global solvability, finite time disappearance of phase and finite time blow-up are considered; similarity solutions and the asymptotic form of the free boundary S(t) as t → ∞ are investigated. For the “inverse” problems which represent (from mathematical point of view) one phase version of binary alloy crystallization problems similarity solutions and local in time solvability theorems are given.