  In this paper we study a system of partial differential equations which
models lithium-ion batteries. The system describes the conservation of Lithium
and conservation of charges in the solid and electrolyte phases, together with
the conservation of energy. The mathematical challenge is due to the fact that
the reaction terms in the system involve the hyperbolic sine function along
with possible degeneracy in one of the high order terms. We obtain a local
existence assertion for the initial boundary problem for the system which
offers insight into how long a battery can last.
