On Maxwell's Equations in an Electromagnetic Field with the Temperature Effect
Hong‐Ming Yin · SIAM Journal on Mathematical Analysis · 1998
This paper deals with Maxwell's equations coupled with a nonlinear heat equation. The system models an induction heating process for a conductive material in which the electrical conductivity strongly depends on the temperature. It is shown that the evolution system has a global weak solution if the electrical conductivity is bounded. For the case of one space dimension, the existence of a global classical solution is established. Moreover, for a quasi-stationary state field it is proved that the temperature will blow up in finite time if the electric conductivity satisfies certain growth conditions.