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Spectral Analysis of a Viscoelasticity Problem

Abstract

An eigenvalue problem associated with small movements of a viscoelastic body fixed on the boundary of a bounded domain is studied. The spectrum of the problem is proved to lie in a vertical strip bounded away from the imaginary axis and to be symmetric about the real axis. The essential spectrum of the problem consists of a finite number of points on the real axis. There are two sequences of complex conjugate eigenvalues condensing toward infinity. Under certain additional conditions, the spectrum that does not lie on the real axis is bounded away from it.

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ACKNOWLEDGMENTS

The author is grateful to Professor N.D. Kopachevskii for discussing this work.

This study was supported by the Ministry of Education and Science of the Russian Federation, project no. 14.Z50.31.0037.

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Correspondence to D. A. Zakora.

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Translated by I. Ruzanova

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Zakora, D.A. Spectral Analysis of a Viscoelasticity Problem. Comput. Math. and Math. Phys. 58, 1761–1774 (2018). https://doi.org/10.1134/S0965542518110179

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  • DOI: https://doi.org/10.1134/S0965542518110179

Keywords:

  • viscoelastic body
  • integro-differential equation
  • spectrum
  • essential spectrum
  • asymptotic behavior of eigenvalues