Block operator matrix techniques for stability properties of hyperbolic equations

TL;DR

This paper develops new criteria for the stability of damped hyperbolic equations in block operator form, with applications to Maxwell's equations, improving regularity and structural assumptions over previous results.

Contribution

It introduces novel stability criteria for hyperbolic equations using block operator matrices, relaxing regularity and structural conditions for Maxwell's equations.

Findings

Established strong and semi-uniform stability criteria.

Applied criteria to Maxwell's equations with minimal regularity assumptions.

Improved upon existing conditions for damping conductivity and domain regularity.

Abstract

Inspired by recent developments in the theory of stability results in the context of certain wave type phenomena, we discuss abstract damped hyperbolic type equations given in a block operator matrix form with regards to asymptotic behaviour of their solutions. Under mild conditions on the operators involved we provide criteria establishing strong or semi-uniform stability. In the particular case of Maxwell's equations, these criteria are implied under mild regularity conditions of the underlying domain causing spatial derivative operators satisfy certain compact embedding conditions and rather minimal assumptions on the damping conductivity. These assumptions improve on both regularity as well as on the structural requirements for the conductivity previously available in the literature.