In this paper, we consider a $2\times 2$ hyperbolic system originates from the
theory of phase dynamics. This one-phase problem can be obtained by using the
Catteneo-Fourier law which is a variant of the standard Fourier law in one dimensional
space. A new classical existence and uniqueness result is established by some
a priori estimates using the characteristic method. The convergence of the solutions
to the one of classical Stefan problems is also obtained.