Geodesic
Solvability of an axisymmetric problem for nonlinear parabolic equation in domains with a non-cylindrical or unknown boundary. I
Čelâbinskij fiziko-matematičeskij žurnal, Tome 5 (2020) no. 1, pp. 44-55

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We prove the regular solvability for problems to quasilinear three-dimensional parabolic equation with the axial symmetry in a non-cylindrical region with a given boundary from the class $W^1_2$ (part I) or an unknown one in general by time (part II). In the second case, the equation describes the processes of phase transitions of a substance from one state to another. The boundary of the transition phase is unknown and is determined together with the solution. Unlike the well-known Stefan's problem, when the latent heat of fusion of a substance is known, here we consider the problem when it is necessary to determine this characteristic, if the volume of the melted substance for a given period is known.
Keywords: Stefan's condition, nonlinear parabolic equation, non-cylindrical domain, compactness theorem. 
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     author = {A. G. Podgaev},
     title = {Solvability of an axisymmetric problem for nonlinear parabolic equation in domains with a non-cylindrical or unknown boundary. {I}},
     journal = {\v{C}el\^abinskij fiziko-matemati\v{c}eskij \v{z}urnal},
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     publisher = {mathdoc},
     volume = {5},
     number = {1},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHFMJ_2020_5_1_a3/}
}
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A. G. Podgaev. Solvability of an axisymmetric problem for nonlinear parabolic equation in domains with a non-cylindrical or unknown boundary. I. Čelâbinskij fiziko-matematičeskij žurnal, Tome 5 (2020) no. 1, pp. 44-55. http://geodesic.mathdoc.fr/item/CHFMJ_2020_5_1_a3/