  In this article we investigate a model for the temperature within a
Lithium-Ion battery. The model takes the form of a parabolic PDE for the
temperature coupled with two elliptic PDE's for the electric potential within
the solid and electrolyte phases. The primary difficulty comes from the
coupling term, which is given by the Butler-Volmer equation. It features an
exponential nonlinearity of both the electric potentials and the reciprocal of
the temperature. Another difficulty arising in the temperature equation are the
gradients of the electric potentials squared showing up on the right-hand side.
Due to the nonlinearity, meaningful estimates for the temperature are currently
not known. In spite of this, our investigation reveals the local existence of
continuous temperature for the Lithium-Ion Battery.
