# Boundary Stabilization of Quasilinear Maxwell Equations

## TL;DR

This paper proves that small initial data for a quasilinear Maxwell system with absorbing boundary conditions in a star-shaped domain lead to unique, globally existing solutions that decay exponentially, using a nonlinear stabilizability approach.

## Contribution

It introduces a novel stabilizability inequality for quasilinear Maxwell equations with absorbing boundaries, enabling proof of global existence and exponential decay of solutions.

## Key findings

- Existence of unique global solutions for small initial data.
- Solutions decay exponentially over time.
- Development of a stabilizability inequality for the system.

## Abstract

We investigate an initial-boundary value problem for a quasilinear nonhomogeneous, anisotropic Maxwell system subject to an absorbing boundary condition of Silver & M\"uller type in a smooth, bounded, strictly star-shaped domain of $\mathbb{R}^{3}$. Imposing usual smallness assumptions in addition to standard regularity and compatibility conditions, a nonlinear stabilizability inequality is obtained by showing nonlinear dissipativity and observability-like estimates enhanced by an intricate regularity analysis. With the stabilizability inequality at hand, the classic nonlinear barrier method is employed to prove that small initial data admit unique classical solutions that exist globally and decay to zero at an exponential rate. Our approach is based on a recently established local well-posedness theory in a class of $\mathcal{H}^{3}$-valued functions.