On the justification of the quasistationary approximation for the Stefan problem

We prove the unique solvability of the one-phase Stefan problem with a small multiplier $\varepsilon$ at the time derivative in the equation on a certain time interval independent of $\varepsilon$ for $\varepsilon\in (0,\varepsilon_0)$. We compare the solution to the Stefan problem with the solution to the Hele-Show problem which describes the process of melting materials with zero specific heat $\varepsilon$ and can be considered as a quasistationary approximation for the Stefan problem. We show that the difference of the solutions has order $\mathcal O(\varepsilon)+\mathcal O(e^{-\frac{ct}{\varepsilon}})$. This provides justification of the quasistationary approximation.