General One-Phase Stefan Problems and Free Boundary Problems for the Heat Equation with Cauchy Data Prescribed on the Free Boundary
Bernard Sherman · SIAM Journal on Applied Mathematics · 1971
where u(x, t) and the free boundary s(t) are to be determined. Here f, g, h, A, p, and a are the data of problem (1.1), with f, g, h, A defined for t > 0 and (p(x) defined for 0 ? x 0 and a > 0 and compatibility and regularity conditions described in Theorem 1 of ? 2. In Theorem 1 we prove existence and uniqueness of the solution of (1.1). If a is the supremum of those t for which (1.1) has a solution, then we prove, in Theorem 1, that either a = oo, or a is finite with s(a-) 0 and s(t) > 0 for t < a, or a is finite, s(t) does not tend to 0 as t T a, and lim inf s'(t) = lim inf ux(s(t), t) = oo as t T a. We describe the Stefan problem (1.1) as general because there are no sign restrictions on f, g, h and (p. We may interpret (1.1) as a problem of heat conduction with melting when