Solvability of an axisymmetric problem for nonlinear parabolic equation in domains with a non-cylindrical or unknown boundary. I

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.