Abstract:
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.
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