Existence and geometry of a free boundary problem for the heat equation (Q1175124)

The following free boundary problem in one phase and one space dimension is considered \[ U_ t=U_{xx}, X(t)<x<s(t), t>0;\quad U(X(t),t)=1, U(s(t),t)=0, U_ x(s(t),t)=-1, t>0. \leqno (1) \] Here \(X(t)\) is a given Lipschitz continuous periodic function and the solution is required to be time periodic. The existence, uniqueness and regularity of the solution is established and the free boundary is shown to be no more complicated than the geometry of the fixed boundary. It is worth noting that if the solution is sufficiently regular, then problem (1) transforms into a Stefan problem for the function \(v=U_ x- 1\) and the corresponding results established for the existence of periodic solutions of Stefan problems could be applied.