Real-time control of the free boundary in a two-phase stefan problem

A two-phase Stefan-problem is considered where the progress of the free boundary is observed by fully automatic real-time controls (thermostats or photo-electric cells). The heat flux at both fixed boundaries can be determined by heaters which respond to the signals of the controls, possibly with a certain time lag. The corresponding mathematical model leads to a two-phase Stefan problem with nonlinear and discontinuous boundary conditions with delays at the fixed boundaries. The problem is transformed into a set-valued fixed point equation, and the existence of a solution is shown with the aid of a theorem due to Bohnenblust-Karlin. The consequence of this result is that a free boundary with the well-known smoothness properties develops under the impact of a fully automatic real-time control via thermostats or photo-electric cells. Some numerical experiments conclude the paper. They indicate that the model is realistic.