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Block operator matrix techniques for stability properties of hyperbolic equations

Marcus Waurick

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.

Block operator matrix techniques for stability properties of hyperbolic equations

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.
Paper Structure (8 sections, 29 theorems, 133 equations)

This paper contains 8 sections, 29 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Changing the variables and the generation theorem for \ref{['eq:absMax']}
  3. A 3-by-3 matrix representation
  4. Strong and semi-uniform stability
  5. Closed range statements for the system operator
  6. Application to Maxwell's equations with partial damping
  7. The proof of the final closed range statement
  8. Conclusion

Key Result

Theorem 2.2

Assume hyp:fne0. Then the operator is m-dissipative and, thus, generates a contraction semi-group on $H$ and on $\overline{\mathop{\mathrm{ran}}\nolimits}(A)$.

Theorems & Definitions (64)

  • Theorem 2.2
  • proof
  • remark 3.2
  • Theorem 3.3: Wa26
  • proof
  • Theorem 4.1: Batty--Duyckaerts, BD08
  • Theorem 4.2: ABLV-Theorem
  • remark 4.4
  • Theorem 4.5
  • Theorem 4.6
  • ...and 54 more