Weighted estimates of a solution to the linear problem connected with one-phase Stefan problem in the case of the specific heat tends to zero

We prove estimates in weighted Hölder norms for a solution to the model linear problem related with the one-phase Stefan problem with a small multiplier $\varepsilon$ at time derivative in the heat equation. These estimates are uniform with respect to parameter $\varepsilon$ and will be significant in the justification of passage to the limit in the one-phase Stefan problem with the specific heat tends to zero.