The aim of this note is to obtain conditions for the existence on a small time interval of a classical solution of two model problems with a free boundary, generated by phase transitions in media with a ``mushy region''. One of these models is the well-known Crank-Gupta problem; the other is called the degenerate Stefan problem. We need to find a function \(u(x,t)\) defined in the time domain \(\Omega(t)\subset \Omega\), \(\forall t\in [0,T]\), and a hypersurface \(\Gamma(t)\subset \Omega\), \(\forall t\in [0,T]\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\), \(n\geq 1\), with a sufficiently smooth boundary \(\partial\Omega\), such that \[ Lu\equiv (\partial/ \partial t- \nabla_ x (A(x,t) \nabla_ x)) u=F \quad \text{in } \Omega(t), \qquad t\in [0,T]. \] We require that \(\Gamma(t)\cap \partial\Omega= \emptyset\) and \(\partial \Omega(t)= \Gamma(t)\cup \partial\Omega\), \(\forall t\in [0,T]\). Also, \[ u|_{t=0}= \varphi_ 0 \quad \text{in } \Omega(0), \qquad \partial u/\partial n_ f=g \quad \text{on } \partial\Omega\times [0,T], \] where \(n_ f\) is the unit normal vector \(\partial\Omega\), and \[ u=0, \qquad K(x)\cdot V_ n= \langle n_ t, A(x,t) \nabla_ x u\rangle \quad \text{on } \Gamma(t), \] where \(n_ t\) is the vector of the unit normal to \(\Gamma(t)\), outward with respect to \(\Omega(t)\), \(\langle \cdot, \cdot\rangle\) is the inner product in \(\mathbb{R}^ n\), and \(V_ n\) the normal velocity of the motion of \(\Gamma(t)\). We assume that the coefficients \(A_{ij}\) of the uniformly elliptic matrix \(A(x,t)\) belong to \(C^{l_ 0,l_ 0/2} (\Omega\times [0,T])\), that \(\Gamma(0)\), \(\partial\Omega\in C^{l_ 0+1}\), where \(l_ 0>28\), and that \(K\in C^{l_ 0-1} (\Omega)\).