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Wave Decay with Singular Damping

Hans Christianson, Emmanuel Schenck, Michael Taylor

Abstract

We consider the stabilization problem on a manifold with boundary for a wave equation with measure-valued linear damping. For a wide class of measures, containing Dirac masses on hypersurfaces as well as measures with fractal support, we establish an abstract energy decay result.

Wave Decay with Singular Damping

Abstract

We consider the stabilization problem on a manifold with boundary for a wave equation with measure-valued linear damping. For a wide class of measures, containing Dirac masses on hypersurfaces as well as measures with fractal support, we establish an abstract energy decay result.

Paper Structure

This paper contains 8 sections, 8 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Measures that multiply $H_{0}^{1}(\Omega)$ to $H^{-1}(\Omega)$
  3. Measures related to hypersurfaces $S\subset M$
  4. Measures with fractal support
  5. Proof of theorem \ref{['T:m-damp-1']}
  6. Decay
  7. Primitive eigenspaces with purely imaginary eigenvalues
  8. Spectral gap and exponential decay

Key Result

Theorem 1

Suppose the linear operator $M$ extends to a bounded linear mapping $M:H_{0}^{1}(\Omega)\to H^{-1}(\Omega)$, and that this mapping is compact. Then $G$ is maximally dissipative, and $(I-G)^{-1}:\mathcal{H}\to\mathcal{H}$ is compact. In particular, $G$ generates a semigroup of contractions $e^{tG}:\m

Theorems & Definitions (15)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Proposition 4
  • proof
  • Remark 5
  • Lemma 6
  • Remark 7
  • Proposition 8
  • proof
  • ...and 5 more