A hyperbolic problem with non-local constraint describing ion-rearrangement in a model for ion-lithium batteries

In this paper we study the Fokker-Plank equation arising in a model which describes the charge and discharge process of ion-lithium batteries. In particular we focus our attention on slow reaction regimes with non-negligible entropic effects, which triggers the mass-splitting transition. At first we prove that the problem is globally well-posed. After that we prove a stability result under some hypothesis of improved regularity and a uniqueness result for the stability under some additional condition of