Weak Solutions to a Nonuniformly Elliptic PDE System in the Harmonic Regime

Abstract

We study the existence of weak solutions to a nonlinear strongly coupled parabolic–elliptic PDEs arising in the heating induction-conduction process of steel hardening. In this setting, our major concern is to consider the case when the electric conductivity is nonuniformly elliptic which, together with a right hand side in L1 in the energy balance equation, yields to a difficult theoretical situation. The existence result gives a weak solution to a similar PDEs system where the energy balance equation has been perturbed by a measure term.