  The life-cycle of electric batteries depends on a complex system of
interacting electrochemical and growth phenomena that produce dendritic
structures during the discharge cycle. We study herein a system of 3 partial
differential equations combining an Allen--Cahn phase-field model (simulating
the dendrite-electrolyte interface) with the Poisson--Nernst--Planck systems
simulating the electrodynamics and leading to the formation of such dendritic
structures. We prove novel existence, uniqueness and stability results for this
system and use it to produce simulations based on a finite element code.
