Free boundary problems for the heat equation in which the moving interface coincides initially with the fixed face

Abstract

A semi-infinite solid at the melting temperature 0 occupies the region 0 ⩽ x < ∞. At the fixed face x = 0 there is prescribed either ( a) the temperature f( t) ⩾ 0 or ( b) the heat flux f( t) directed into the solid, where f( t) is not identically zero in the vicinity of t = 0. There is a free boundary s( t) separating the liquid phase from the solid phase, and a temperature distribution u( x, t) in the liquid phase. We prove existence, uniqueness, and continuous dependence theorems for problems ( a) and ( b). Let F( t) be the integral, from 0 to t, of f(τ). Then we prove that for problem ( a) s( t) behaves, in the vicinity of t = 0, essentially like the square root of F( t) while for problem ( b) s( t) behaves essentially like F( t). We obtain also upper bounds for s′( t) for both problems. The existence theorems are proved as follows: the region 0 ⩽ x ⩽ a is taken to be liquid at the melting temperature and the region a < x < ∞ solid at the melting temperature. Problems ( a) and ( b) are known to have solutions u a and s a in this case which are bounded, equicontinuous families. By Arzela's theorem uniformly convergent subsequences can be extracted. These subsequences converge to the solutions of ( a) and ( b) for the case a = 0.