A free boundary value problem for the heat equation
Differentiation is denoted by a subscript whether it is a partial derivative of a function of two variables or an ordinary derivative of a function of one variable. This problem with A = 0 has been discussed by several authors, see for example [1, 2, 6], but thus far no existence proof has been established**. The problem describes the physical phenomena of evaporation, fusion, sublimation, etc. For example with B = 0, (1) could refer to the following situation. A long metal rod insulated at the sides has begun to melt at one end (x = 0). The layer of liquid metal is A units deep and has some initial temperature distribution, F(x). The critical temperature Tc is the melting point of the metal. At x = 0 heat is applied to the rod at a known rate proportional