Advertisement

On the type ofCo-semigroup associated with the abstract linear viscoelastic system

  • Kangsheng Liu
  • Zhuangyi Liu
Original Papers

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.

Keywords

Mathematical Method Growth Property Spectrum Point Linear Viscoelasticity Abstract Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [BS]
    I. N. Bronshtein and K. A. Semendyayev,Handbook of Mathematics, Van Nostrand Reinhold Co. 1985.Google Scholar
  2. [Ch]
    R. M. Christensen,Theory of Viscoelasticity, An Introduction, 2nd ed., Academic Press Inc. 1982.Google Scholar
  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.Google Scholar
  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.Google Scholar
  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

Copyright information

© Birkhäuser Verlag 1996

Authors and Affiliations

  • Kangsheng Liu
    • 1
  • Zhuangyi Liu
    • 2
  1. 1.Center for Mathematical SciencesZhejiang UniversityHangzhouChina
  2. 2.Dept of Mathematics and StatisticsUniversity of MinnesotaDuluthUSA

Personalised recommendations