On the type ofC o-semigroup associated with the abstract linear viscoelastic system

Abstract

In this paper, we give an estimate for the type of semigroup associated with an abstract equation of linear viscoelasticity when the memory kernel decays exponentially. In particular, when the kernel is of Maxwell type, we prove that the spectrum determined growth property holds. Moreover, the type of the semigroup is explicitly expressed by a formula which depends on the parameters of the kernel and the minimum spectrum point of the corresponding elastic operator.

This is a preview of subscription content, access via your institution.

References

  1. [BS]

    I. N. Bronshtein and K. A. Semendyayev,Handbook of Mathematics, Van Nostrand Reinhold Co. 1985.

  2. [Ch]

    R. M. Christensen,Theory of Viscoelasticity, An Introduction, 2nd ed., Academic Press Inc. 1982.

  3. [D1]

    C. M. Dafermos,An abstract Volterra equation with applications to linear Viscoelasticity, J. Diff. Eq. 7, 554–569 (1970).

    Google Scholar 

  4. [D2]

    C. M. Dafermos,Asymptotic stability in viscoelasticity, Arch. Rat. Mech. Anal.37, 297–308 (1970).

    Google Scholar 

  5. [FI1]

    R. H. Fabiano and K. Ito,Semigroup theory in linear viscoelasticity: weakly and strongly singular kernels, International Series of Numerical Mathematics, Vol. 91, Birkhäuser, 1989, pp. 109–121.

  6. [FI2]

    R. H. Fabiano and K. Ito,Semigroup theory and numerical approximation for equations arising in linear viscoelasticity, SIAM J. Math. Anal.21(2), 374–393 (1990).

    Google Scholar 

  7. [Hu]

    F. L. Huang,Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. of Diff. Eqs.1(1), 43–56 (1985).

    Google Scholar 

  8. [L]

    J. Lagnese,Boundary Stabilization of Thin Plates, Vol. 10 ofSIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia 1989.

    Google Scholar 

  9. [LZ]

    Z. Liu and S. Zheng,Uniform exponential stability of semigroups associated with approximations of linear viscoelasticity, J. Math. Systems, Estimations, and Control, to appear.

  10. [Rl]

    M. Renardy,On the type of certain C o-semigroups, Comm. Part. Diff. Eq.18, 1299–1307 (1993).

    Google Scholar 

  11. [R2]

    M. Renardy,On linear stability of hyperbolic PDEs and viscoelastic flows. Z. angew Math. Phys.45, 854–865 (1994).

    Google Scholar 

  12. [Pa]

    A. Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York 1983.

    Google Scholar 

  13. [Pr]

    J. Prüss,On the spectrum of C o-semigroups. Trans. Amer. Math. Soc.284, 847–857 (1984).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

Supported partially by the Chinese Natural Science Foundation.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Liu, K., Liu, Z. On the type ofC o-semigroup associated with the abstract linear viscoelastic system. Z. angew. Math. Phys. 47, 1–15 (1996). https://doi.org/10.1007/BF00917570

Download citation

Keywords

  • Mathematical Method
  • Growth Property
  • Spectrum Point
  • Linear Viscoelasticity
  • Abstract Equation