Global existence for a non-local mean curvature flow as a limit of a parabolic-elliptic phase transition model

We consider a free boundary problem where the velocity depends on the mean curvature and on some non-local term. This problem arises as the singular limit of a reaction-diffusion system which describes the microphase separation of diblock copolymers. The interface may present singularities in finite time. This leads us to consider weak solutions on an arbitrary time interval and to prove the global-in-time convergence of solutions of the reaction-diffusion system.