Exponential Stability to a Contact Problem of Partially Viscoelastic Materials

  • Jaime E. Muñoz Rivera
  • Higidio Portillo Oquendo


In this paper we study models for contact problems of materials consisting of an elastic part (without memory) and a viscoelastic part, where the dissipation given by the memory is effective. We show that the solution of the corresponding viscoelastic equation decays exponentially to zero as time goes to infinity, provided the relaxation function also decays exponentially, no matter how small is the dissipative part of the material.

contact problem exponential decay materials with memory Signorini's problem 


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  1. 1.
    C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30(1992) 1024–1065.Google Scholar
  2. 2.
    M.I.M. Copetti and C.M. Elliot, A one dimensional quasi-static contact problem in linear thermoelasticity. Eur. J. Appl. Math. 4(1993) 151–174.Google Scholar
  3. 3.
    C.M. Dafermos, An abstract Volterra equation with application to linear viscoelasticity. J. Differential Equations 7(1970) 554–589.Google Scholar
  4. 4.
    R.P. Gilbert, P. Shi and M. Shillor, A quasistatic contact problem in linear thermoelasticity. Rend. Mat. 10(1990) 785–808.Google Scholar
  5. 5.
    J.M. Greenberg and Li Tatsien, The effect of the boundary damping for the quasilinear wave equation. J. Differential Equations 52(1) (1984) 66–75.Google Scholar
  6. 6.
    M.A. Horn and I. Lasiecka, Uniform decay of weak solutions to a Von Karman plate with nonlinear boundary dissipation. Differential Integral Equations 7(4) (1994) 885–908.Google Scholar
  7. 7.
    G. Gripenberg, S. Londen and O. Staffans, Volterra Integral and Functional Equations. Encyclopedia Math. Appl. 34, Cambridge Univ. Press (1990).Google Scholar
  8. 8.
    F.A. Khodja, A. Benabdallah and D. Teniou, Stabilisation frontière et interne du système de la Thermoélasticité. Reprint de l'équipe de Mathématiques de Besançon 95/32.Google Scholar
  9. 9.
    J.U. Kim, A boundary thin obstacle problem for a wave equation. Comm. Partial Differential Equations 14(8–9) (1989) 1011–1026.Google Scholar
  10. 10.
    V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. 69(1990) 33–54.Google Scholar
  11. 11.
    V. Komornik, Rapid boundary stabilization of the wave equation. SIAM J. Control Optim. 29(1991) 197–208.Google Scholar
  12. 12.
    J.E. Lagnese, Asymptotic Energy Estimates for Kirchhoff Plates Subject to Weak Viscoelastic Damping. Internat. Ser. Numer. Math. 91, Birkhäuser, Basel (1989).Google Scholar
  13. 13.
    I. Lasiecka and R. Trigianni, Exact controllability and uniform stabilization of Euler–Bernoulli equations with boundary control only in Δ|Σ. Boll. Un. Mat. Ital. 7(1991) 665–702.Google Scholar
  14. 14.
    I. Lasiecka, Global uniform decay rates for the solution to the wave equation with nonlinear boundary conditions. Appl. Anal. 47(1992) 191–212.Google Scholar
  15. 15.
    I. Lasiecka, Exponential decay rates for the solutions of Euler–Bernoulli equations with boundary dissipation ocurring in the moments only. J. Differential Equations 95(1992) 169–182.Google Scholar
  16. 16.
    J.L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Masson, Paris (1988).Google Scholar
  17. 17.
    K. Liu and Z. Liu, Exponential decay of the energy of the Euler–Bernoulli beam with locally distributed Kelvin–Voigt damping. SIAM J. Control Optim. 36(3) (1998) 1086–1098.Google Scholar
  18. 18.
    J.E. Muñoz Rivera and J. Sobrinho, Existence and uniform rates of decay to contact problem in viscoelasticity. Appl. Anal. 67(3–4) (1998) 175–200.Google Scholar
  19. 19.
    M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation. Math. Ann. 305(1996) 403–417.Google Scholar
  20. 20.
    K. Ono, A stretched string equation with a boundary dissipation. Kyushu J. Math. V 28(2) (1994) 265–281.Google Scholar
  21. 21.
    J.P. Puel and M. Tucsnak, Boundary stabilization for the Von Kármán equation. SIAMJ. Control Otim. 33(1) (1995) 255–273.Google Scholar
  22. 22.
    J.P. Puel and M. Tucsnak, Global existence for the full von Kármán system. Appl. Math. Optim. 34(1996) 139–160.Google Scholar
  23. 23.
    J. Ralston, Solution of the wave equation with localized energy. Comm. Pure Appl. Math. 22(1969) 807–823.Google Scholar
  24. 24.
    J. Ralston, Gaussian beams and the propagation of singularities. In:W. Littman (ed), Studies in PDEs, MAA Studies in Math. 23 (1982), pp. 206–248.Google Scholar
  25. 25.
    M. Renardy, On the type of certain C0-semigroups. Comm. Partial Differential Equations 18(1993) 1299–1307.Google Scholar
  26. 26.
    A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potencial. J. Math. Pures Appl. 71(1992) 455–467.Google Scholar
  27. 27.
    Shen Weixi and Zheng Songmu, Global smooth solution to the system of one dimensional thermoelasticity with dissipation boundary condition. Chin. Ann. of Math. 7B(3) (1986) 303–317.Google Scholar
  28. 28.
    P. Shi and M. Shillor, Existence of a solution to the ndimensional problem of thermoelastic contact. Comm. Partial Differential Equations 17(9–10) (1992) 1597–1618.Google Scholar
  29. 29.
    P. Shi and M. Shillor, Uniqueness and stability of the solution to a thermoelastic contact problem. Eur. J. Appl. Math. 1(1990) 371–387.Google Scholar
  30. 30.
    P. Shi and M. Shillor, A quasistatic contact problem in thermoelasticity with a radiation condition for the temperature. J. Math. Anal. Appl. 172(1) (1993) 147–165.Google Scholar
  31. 31.
    P. Shi, M. Shillor and X.L. Zou, Numerical solution to the one dimensional problems of thermoelastic contact. Comput. Math. Appl. 22(10) (1991) 65–78.Google Scholar
  32. 32.
    E. Zuazua, Exponential decay for the semilinear wave equation with locally distribuited damping. Comm. Partial Differential Equations 15(1990) 205–235.Google Scholar
  33. 33.
    E. Zuazua, Uniform stabilization of the wave equation by nonlinear wave equation boundary feedback. SIAM J. Control Optim. 28(1990) 466–477.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Jaime E. Muñoz Rivera
    • 1
    • 2
  • Higidio Portillo Oquendo
    • 1
  1. 1.Department of Applied MathematicNational Laboratory for Scientific ComputationPetrópolis, RJBrazil
  2. 2.IMFederal University of Rio de JaneiroBrazil

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