A problem with a free boundary for nonlinear equation with a change in the direction of evolution

We prove the regular solvability of a problem with axial symmetry for a quasilinear multidimensional equation with a change in the direction of parabolicity and an unknown type change boundary from the class $W_2^1$. On this unknown boundary of the change in the direction of evolution, a condition similar to the Stefan condition is set, in which the constant (playing, in the case of the Stefan problem for the heat equation, the role of the latent specific heat of melting of the substance) is also unknown.