Normal trace inequalities and decay of solutions to the nonlinear Maxwell system with absorbing boundary

TL;DR

This paper investigates the nonlinear Maxwell system with absorbing boundaries, establishing global existence and exponential decay of solutions under small data conditions, using novel trace estimates and regularity analysis.

Contribution

Introduces new trace estimates and analytical techniques for studying decay in nonlinear Maxwell systems with boundary absorption.

Findings

Solutions exist globally for small initial data.

Solutions decay exponentially over time.

Enhanced results for the linear autonomous case.

Abstract

We study the quasilinear Maxwell system with a strictly positive, state dependent boundary conductivity. For small data we show that the solution exists for all times and decays exponentially to $0$. As in related literature we assume a nontrapping condition. Our approach relies on a new trace estimate for the corresponding non-autonomous linear problem, an observability-type estimate, and a detailed regularity analysis. The results are improved in the linear autonomous case, using properties of the Helmholtz decomposition in Sobolev spaces of (small) negative order.