We study the two-phase Stefan problem that models heat propagation and phase transitions in a material with two distinct phases, such as
water and ice. For this problem, we introduce a notion of viscosity
solutions that allows for an appearance of the so-called mushy region. We prove a comparison principle and use this result to establish well-posedness of the viscosity solutions. As a corollary, we show that the viscosity solutions and the weak solutions defined in the sense of distributions coincide.