  In this paper we study the initial boundary value problem for the system
$\mbox{div}(\sigma(u)\nabla\varphi)=0,\ \ u_t-\Delta
u=\sigma(u)|\nabla\varphi|^2$. This problem is known as the thermistor problem
which models the electrical heating of conductors. Our assumptions on
$\sigma(u)$ leave open the possibility that
$\liminf_{u\rightarrow\infty}\sigma(u)=0$, while
$\limsup_{u\rightarrow\infty}\sigma(u)$ is large. This means that $\sigma(u)$
can oscillate wildly between $0$ and a large positive number as $u\rightarrow
\infty$. Thus our degeneracy is fundamentally different from the one that is
present in porous medium type of equations. We obtain a weak solution $(u,
\varphi)$ with $|\nabla \varphi|, |\nabla u|\in L^\infty$ by first establishing
a uniform upper bound for $e^{\varepsilon u}$ for some small $\varepsilon$.
This leads to an inequality in $\nabla\varphi$, from whence follows the
regularity result. This approach enables us to avoid first proving the
H\"{o}lder continuity of $\varphi$ in the space variables, which would have
required that the elliptic coefficient $\sigma(u)$ be an $A_2$ weight. As it is
known, the latter implies that $\ln\sigma(u)$ is "nearly bounded".
