In this paper, we shall establish the well-posedness of a mathematical model for a special class of electrochemical power device – lithium-ion battery. The underlying partial differential equations in the model involve a (mix and fully) coupled system of quasi-linear elliptic and parabolic equations. By exploring some special structure, we are able to adopt the well-known Nash-Moser- DeGiorgi boot strap to establish suitable a priori supremum estimates for the electric potentials. Using the supremum estimates, we apply the Leray-Schauder theory to establish the existence and uniqueness of a subsystem of elliptic equations that describe the electric potentials in the model. We then employ a Schauder fix point theorem to obtain the local (in time) existence for the whole model. We also consider the global existence of a modified 1-d governing system under additional assumptions. In particular, we are able to derive uniform a priori estimates depending only on the existence time $T$, including the supremum estimates for electric potentials and growth and decay estimates for the concentration $c$. Using the uniform estimates, we prove that the modified system has a solution for all time $t>0$.
"On the Well-posedness of a Mathematical Model for Lithium-Ion Battery Systems." Methods Appl. Anal. 13 (3) 275 - 298, September 2006.