Milan Journal of Mathematics

, 77:333 | Cite as

Stability and exponential stability in linear viscoelasticity

  • Vittorino PataEmail author


In this survey paper, we discuss the decay properties of the semigroup generated by a linear integro-differential equation in a Hilbert space, which is an abstract version of the equation
$${\partial_{tt}}u(t) - \Delta u(t) + {\int_0^\infty} \mu(s) \Delta u(t - s) {\rm{d}}s = 0$$
describing the dynamics of linearly viscoelastic bodies.

Mathematics Subject Classification (2000)

35B40 45K05 45M10 47D06 


Linear viscoelasticity memory kernels contraction semigroups stability exponential stability 


  1. 1.
    Chepyzhov V.V., Mainini E., Pata V.: Stability of abstract linear semigroups arising from heat conduction with memory. Asymptot. Anal. 46, 251–273 (2006)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Chepyzhov V.V., Pata V.: Some remarks on stability of semigroups arising from linear viscoelasticity. Asymptot. Anal. 50, 269–291 (2006)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Conti M., Gatti S., Pata V.: Uniform decay properties of linear Volterra integro-differential equations. Math. Models Methods Appl. Sci. 18, 1–21 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Curtain R.F., Zwart H.J.: An introduction to infinite-dimensional linear system theory. Springer, New York (1995)Google Scholar
  5. 5.
    Dafermos C.M.: Asymptotic stability in viscoelasticity. Arch. Rational Mech. Anal. 37, 297–308 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    C.M. Dafermos, Contraction semigroups and trend to equilibrium in continuum mechanics, in “Applications of Methods of Functional Analysis to Problems in Mechanics” (P. Germain and B. Nayroles, Eds.), pp.295–306, Lecture Notes in Mathematics no.503, Springer-Verlag, Berlin-New York, 1976.Google Scholar
  7. 7.
    Fabrizio M., Lazzari B.: On the existence and asymptotic stability of solutions for linear viscoelastic solids. Arch. Rational Mech. Anal. 116, 139–152 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    M. Fabrizio, A. Morro, Mathematical problems in linear viscoelasticity, SIAM Studies in Applied Mathematics no.12, SIAM, Philadelphia, 1992.Google Scholar
  9. 9.
    Fabrizio M., Polidoro S.: Asymptotic decay for some differential systems with fading memory. Appl. Anal. 81, 1245–1264 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    C. Giorgi, B. Lazzari, Uniqueness and stability in linear viscoelasticity: some counterexamples, in “Waves and stability in continuous media (Sorrento, 1989)” pp.146–153, Ser. Adv. Math. Appl. Sci. no.4, World Sci. Publishing, River Edge, NJ, 1991.Google Scholar
  11. 11.
    Giorgi C., Lazzari B.: On the stability for linear viscoelastic solids. Quart. Appl. Math. 55, 659–675 (1997)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Giorgi C., Muñoz Rivera J.E., Pata V.: Global attractors for a semilinear hyperbolic equation in viscoelasticity. J. Math. Anal. Appl. 260, 83–99 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Grasselli M., Muñoz Rivera J.E., Pata V.: On the decay of the linear thermoelastic plate with memory. J. Math. Anal. Appl. 309, 1–14 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    M. Grasselli, V. Pata, Uniform attractors of nonautonomous systems with memory, in “Evolution Equations, Semigroups and Functional Analysis” (A. Lorenzi and B. Ruf, Eds.), pp.155–178, Progr. Nonlinear Differential Equations Appl. no.50, Birkhäuser, Boston, 2002.Google Scholar
  15. 15.
    Liu Z., Zheng S.: On the exponential stability of linear viscoelasticity and thermoviscoelasticity. Quart. Appl. Math. 54, 21–31 (1996)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Z. Liu, S. Zheng, Semigroups associated with dissipative systems, Chapman & Hall/CRC Research Notes in Mathematics no.398, Chapman & Hall/CRC, Boca Raton, FL, 1999.Google Scholar
  17. 17.
    Muñoz Rivera J.E.: Asymptotic behaviour in linear viscoelasticity. Quart. Appl. Math. 52, 629–648 (1994)zbMATHGoogle Scholar
  18. 18.
    Pata V.: Exponential stability in linear viscoelasticity. Quart. Appl. Math. 64, 499–513 (2006)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Pata V., Zucchi A.: Attractors for a damped hyperbolic equation with linear memory. Adv. Math. Sci. Appl. 11, 505–529 (2001)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Pazy A.: Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York (1983)zbMATHGoogle Scholar
  21. 21.
    Prüss J.: On the spectrum of C 0-semigroups. Trans. Amer. Math. Soc. 284, 847–857 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Renardy M., Hrusa W.J., Nohel J.A.: Mathematical problems in viscoelasticity. Harlow John Wiley & Sons, Inc., New York (1987)zbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Politecnico di Milano - Dipartimento di Matematica “F.Brioschi”MilanoItaly

Personalised recommendations