  We study solutions to a variational equation that models heat control on the
boundary. This problem can be thought of as the two phase parabolic Signorini
problem. We show that when the solution has a sign on the boundary, the study
of the free boundary can be reduced to the study of the free boundary in the
parabolic Signorini problem. We show that locally there is always a sign on the
boundary; therefore, the optimal regularity of the solutions as well as the
study of the free boundary is reduced to the one phase parabolic Signorini
problem.
