Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion

We consider an inverse 2D free-boundary elliptic-parabolic problem, modeling the transient regime of a magnetically conned plasma in a non ideal Stellarator device. The inverse nature of the problem comes from the fact that the associated Grad-Shafranov equation involves some unknown nonlinear terms which must be determined by the current-carrying Stellarator condition. One of the main diculties for the mathematical approach is due to the fact that these nonlinear terms are neither a monotone function and are nor a Lipschitz function either. Nevertheless, we can introduce an auxiliary problem for which the solution of our original problem is a supersolution. On the other hand, we can to obtain a subsolution for this new problem. Finally, by applying the comparison principle for quasi-linear problems, we obtain some appropriate estimates on the location and size of the plasma region.