Properties of the free boundary in the parabolic problems with hysteresis

We consider the problem with a free boundary described by a one- dimensional heat equation with a discontinuous right part generated by a hysteresis-type operator.
It is established that for transversal initial data from the space $W_q^{2-2/q}$, $q>3$, the problem is solvable in the space $W_q^{2,1}$, and the free (interphase) boundaries are defined by monotone Hölder curves with exponent $1/2$. It is also shown that if the initial data belong to the space $W_\infty^2$, the interphase boundaries satisfy the Lipschitz condition.
The paper is based on results obtained jointly with N.N. Uraltseva.