On the classical solution of the multidimensional Stefan problem for quasilinear parabolic equations

In this paper the author proves a theorem on the existence of a classical solution of the Stefan problem for the equation
$$ D_t\theta=\sum^n_{i,j=1}D_i[a_{ij}(x,t,\theta)D_j\theta]+f(x,t,\theta,D\theta) $$
on a small time interval.
The solution is obtained as a limit as $\varepsilon\to0$ of solutions of auxiliary “regularized” problems. Estimates for solutions of the auxiliary problems are established that do not depend on $\varepsilon$. These estimates permit one to say something about the compactness of the family of solutions in the space $C^{2,1}$.
Bibliography: 13 titles.