Far Eastern Mathematical Journal

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Global three-dimensional solvability the axisimmetric Stefan problem for quasilinear equation

Podgaev A. G., Prudnikov V. Ya., Kulesh T. D.

2022, issue 1, P. 61-75
DOI: https://doi.org/10.47910/FEMJ202206

We prove results related to the study of the solvability of a problem with an unknown boundary by compactness methods. Relative compactness theorems are used, which were obtained in previous publications, adapted to the study of problems like the Stefan problem with an unknown boundary. In previous papers, for the equation considered here, we studied the initial-boundary problem in a non-cylindrical domain with a given curvilinear boundary of class $W^1_2$ and the problem for which, under the condition on the unknown boundary, the coefficient latent specific heat of fusion (in contrast to the Stefan problem, considered given here) was an unknown quantity. Therefore, in some places calculations will be omitted that almost completely coincide with those set out in the works listed below. The proposed technique can be applied in more general situations: more phase transition boundaries, or more complex nonlinearities. As a result, global over time, the regular solvability of a single-phase axisymmetric Stefan problem for a quasilinear three-dimensional parabolic equation with unknown boundary from the class $W^1_4$, is proved.

Stefan problem, relative compactness, non-cylindrical domain, unknown boundary

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