Solvability of a Family of Nonlinear Degenerate Parabolic Mixed Equations

In this paper, we show an abstract setting for analyzing the solvability of a class of nonlinear mixed degenerate parabolic equations. Specifically, we combine the well-known theory for stationary mixed problems and the theory for nonlinear degenerate parabolic equations, to obtain sufficient conditions inspired in real-world application problems to guarantee the existence and uniqueness of solution. Furthermore, we illustrate some applications of the abstract framework through particular problems arising from the field of physical electromagnetic sciences. Initially, we introduce two formulations based on a time primitive of the electric field, encompassing both local and nonlocal source conditions. Finally, we explore a nonlinear axisymmetric eddy current model employed in the context of induction furnaces.