  We study a model for induction hardening of steel. The related differential
system consists of a time domain vector potential formulation of the Maxwell's
equations coupled with an internal energy balance and an ODE for the volume
fraction of {\sl austenite}, the high temperature phase in steel. We first
solve the initial boundary value problem associated by means of a Schauder
fixed point argument coupled with suitable a-priori estimates and regularity
results. Moreover, we prove a stability estimate entailing, in particular,
uniqueness of solutions for our Cauchy problem. We conclude with some finite
element simulations for the coupled system.
