Uniform BMO estimate of parabolic equations and global well-posedness of the thermistor problem

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.