One-dimensional parabolic problem arising in the study of some free boundary problems

The paper is concerned with one-dimensional parabolic problem in the domain bounded by straight lines $x=0$ and $x=kt$, $k>0$, $(x,t)\in\mathbb R^2$, with the Neumann boundary condition on the line $x=0$ and with dynamic boundary condition on the lne $x=kt$. For the solution of this problem coercive estimate in a weighted Hölder norm is obtained. It is shown that this estimate can be useful for the analysis of parabolic free boundary problems.