We prove the existence of a weak solution for a two-phase continuous casting Stefan problem with a general monotone nonlinear cooling condition. We establish a sufficient condition for stability, which yields uniqueness and comparison results for the evolutionary and the steady- state solutions. We also discuss the asymptotic behaviour as t←∞ of the corresponding temperatures and enthalpies.

In this paper we consider a superconductor free boundary problem. Under isothermal conditions, a superconductor material (of ‘type I’) will develop two phases separated by a sharp interface Γ(t). In the ‘normal’ conducting phase the magnetic field is divergence free and satisfies the heat equation, whereas on the interface Γ(t), curl , where n is the normal and Vn is the velocity of Γ(t) in the direction of n; further, (constant) on Γ(t). Existence and uniqueness of a classical solution locally in time are established by Newton's iteration method under assumptions which enable us to reduce the 3-dimensional problem to a problem depending on essentially two space variables.

In this paper a one-dimensional parabolic variational inequality which typically arises in option pricing of fixed rate mortgage loan is studied. The main goal is to study the properties of the free boundary. The monotonicity and $C^\infty$ smoothness of free boundary are proved and its behavior near expiry is considered as well.

A free boundary problem for a parabolic system arising from the mathematical theory of combustion will be considered in the one dimensional case. The existence and uniqueness of the classical solution locally in time will be obtained by the use of a fixed point theorem. Also the existence of the classical solution globally in time and a convergence result with respect to a parameter $\lambda$ will be proved under some reasonable assumptions.

A parabolic system with an unknown boundary is considered. The boundary condition at the unknown boundary is concerned with its mean curvature. The physical prototype is superconductivity of type I. Existence of classical solutions is proved by the use of fixed-point theorem, uniqueness is obtained as well.

In this paper, we prove that the Hele–Shaw problem with kinetic condition and surface tension is the limit case of the supercooled Stefan problem in the classical sense when specific heat ε goes to zero. The method is the use of a fixed-point theorem; the key is to construct a workable function space. The main feature is to obtain the existence and the uniform estimates with respect to ε > 0 at the same time for the solutions of the supercooled Stefan problem. For the sake of simplicity, we only consider the case of one phase, although the method used here is also applicable in the case of two phases.