Geodesic
On the classical solution of the multidimensional Stefan problem for quasilinear parabolic equations
Sbornik. Mathematics, Tome 40 (1981) no. 2, pp. 157-178

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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. 
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     author = {A. M. Meirmanov},
     title = {On the classical solution of the multidimensional {Stefan} problem for quasilinear parabolic equations},
     journal = {Sbornik. Mathematics},
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     number = {2},
     year = {1981},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1981_40_2_a2/}
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A. M. Meirmanov. On the classical solution of the multidimensional Stefan problem for quasilinear parabolic equations. Sbornik. Mathematics, Tome 40 (1981) no. 2, pp. 157-178. http://geodesic.mathdoc.fr/item/SM_1981_40_2_a2/