Existence of global weak solutions for a class of quasilinear equations describing Joule's heating

Existence of a weak solution to a nonlinear induction hardening problem with Leblond–Devaux model for a steel workpiece

“…The global solvability of the system was established in some weak sense, and the regularity of the solution was proved using the fixed point theorem. The papers [4] and [5] considered degenerate nonlinear problems describing the Joule heating in terms of unknown electric field and temperature. The existence of global-in-time weak solutions to those problems was shown via the Galerkin method.…”