A doubly nonlinear evolution problem related to a model for microwave heating

TL;DR

This paper proves existence and uniqueness of solutions for a complex doubly nonlinear parabolic PDE model derived from microwave heating, involving maximal monotone operators in the bulk and boundary conditions.

Contribution

It introduces a general framework allowing maximal monotone nonlinearities in both the PDE and boundary, establishing existence and uniqueness results.

Findings

Existence and uniqueness of solutions proved.

Applicable to a broad class of nonlinearities.

Mathematically rigorous treatment of the boundary condition nonlinearities.

Abstract

This paper is concerned with the existence and uniqueness of the solution to a doubly nonlinear parabolic problem which arises directly from a circuit model of microwave heating. Beyond the relevance from a physical point of view, the problem is very interesting also in a mathematical approach: in fact, it consists of a nonlinear partial differential equation with a further nonlinearity in the boundary condition. Actually, we are going to prove a general result: the two nonlinearities are allowed to be maximal monotone operators and then an existence result will be shown for the resulting problem.