Well-Posedness and Exponential Stability of Nonlinear Maxwell Equations for Dispersive Materials with Interface

TL;DR

This paper establishes well-posedness and exponential stability for nonlinear Maxwell equations with dispersive media and interfaces, addressing complex integro-differential models of electromagnetic fields in optical media.

Contribution

It introduces a framework for analyzing nonlinear Maxwell systems with time-delayed polarization, proving well-posedness and stability under specific material conditions.

Findings

Proved well-posedness of the nonlinear Maxwell system

Established exponential stability under certain material conditions

Handled integro-differential equations with interface effects

Abstract

In this paper we consider an abstract Cauchy problem for a Maxwell system modelling electromagnetic fields in the presence of an interface between optical media. The electric polarization is in general time-delayed and nonlinear, turning the macroscopic Maxwell equations into a system of nonlinear integro-differential equations. Within the framework of evolutionary equations, we obtain well-posedness in function spaces exponentially weighted in time and of different spatial regularity and formulate various conditions on the material functions, leading to exponential stability on a bounded domain.