  Global well-posedness of the time-dependent (degenerate) thermistor problem
remains open for many years. In this paper, we solve the problem by
establishing a uniform-in-time BMO estimate of inhomogeneous parabolic
equations. Applying this estimate to the temperature equation, we derive a BMO
bound of the temperature uniform with respect to time, which implies that the
electric conductivity is a $A_2$ weight. The H\"{o}lder continuity of the
electric potential is then proved by applying the De Giorgi--Nash--Moser
estimate for degenerate elliptic equations with $A_2$ coefficient. Uniqueness
of solution is proved based on the established regularity of the weak solution.
Our results also imply the existence of a global classical solution when the
initial and boundary data are smooth.
