Skip to main content
SpringerLink
Log in

Uniform Decay Rates for Viscoelastic Dissipative Systems

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

In this paper, we consider the wave equation with both weak frictional damping and viscoelastic damping acting simultaneously and complementarily in the domain. We establish an explicit and general decay rate result, using some properties of the convex functions. Our result is obtained without imposing any restrictive growth assumption on the frictional damping term and strongly weakening the usual assumptions on the relaxation function.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alabau-Boussouira F. On convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems. Appl Math Optim. 2005;51:61–105.

    Article  MathSciNet  MATH  Google Scholar 

  2. Alabau-Boussouira F, Cannarsa P, Sforza D. Decay estimates for the second order evaluation equation with memory. J Funct Anal. 2008;245:1342–1372.

    Article  MathSciNet  Google Scholar 

  3. Arnold VI. Mathematical methods of classical mechanics. New York: Springer; 1989.

    Book  Google Scholar 

  4. Barreto R, Lapa EC, Munoz Rivera JE. Decay rates for viscoelastic plates with memory. J Elasticity. 1996;44(1):61–87.

    Article  MathSciNet  MATH  Google Scholar 

  5. Barreto R, Munoz Rivera JE. Uniform rates of decay in nonlinear viscoelasticity for polynomial decaying kernels. Appl Anal. 1996;60:263–283.

    MathSciNet  Google Scholar 

  6. Benaissa A, Mimouni S. Energy decay of solutions of a wave equation of p-Laplacian type with a weakly nonlinear dissipation. J Inequal Pure Appl Math. 2006;7(1):8. Article 15.

    MathSciNet  Google Scholar 

  7. Berrimi S, Messaoudi SA. Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping. Elect J Diff Eqns. 2004;2004(88):1–10.

    Google Scholar 

  8. Berrimi S, Messaoudi SA. Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal. 2006;64:2314–2331.

    Article  MathSciNet  MATH  Google Scholar 

  9. Cabanillas EL, Munoz Rivera JE. Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomial decaying kernels. Comm Math Phys. 1996;177:583–602.

    Article  MathSciNet  MATH  Google Scholar 

  10. Cannarsa P, Sforza D. Integro-differential equations of hyperbolic type with positive definite kernels. J Diff Equat. 2011;250:4289–4335.

    Article  MathSciNet  MATH  Google Scholar 

  11. Cavalcanti MM, Domingos Cavalcanti VN, Lasiecka I. Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction. J Diff Equat. 2007;236:407–459.

    Article  MathSciNet  MATH  Google Scholar 

  12. Cavalcanti MM, Domingos Cavalcanti VN, Martinez P. General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal. 2008;68(1):177–193.

    Article  MathSciNet  MATH  Google Scholar 

  13. Cavalcanti MM, Domingos Cavalcanti VN, Prates Filho JS, Soriano JA. Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differ Integr Equ. 2001;14(1):85–116.

    MathSciNet  MATH  Google Scholar 

  14. Cavalcanti MM, Domingos Cavalcanti VN, Soriano JA. Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Elect J Diff Eqns. 2002;2002(44):1–14.

    Google Scholar 

  15. Cavalcanti MM, Oquendo HP. Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 2003;42(4):1310–1324.

    Article  MathSciNet  MATH  Google Scholar 

  16. Dafermos CM. Asymptotic stability in viscoelasticity. Arch Ration Mech Anal. 1970;37:297–308.

    Article  MathSciNet  MATH  Google Scholar 

  17. Dafermos CM. On abstract Volterra equations with applications to linear viscoelasticity. J Differ Equat. 1970;7:554–569.

    Article  MathSciNet  MATH  Google Scholar 

  18. Fabrizio M, Polidoro S. Asymptotic decay for some differential systems with fading memory. Appl Anal. 2002;81:1245–1264.

    Article  MathSciNet  MATH  Google Scholar 

  19. Komornik V. Decay estimates for the wave equation with internal damping. Int Ser Numer Math. 1994;118: 253–266.

    MathSciNet  Google Scholar 

  20. Komornik V, Zuazua E. A direct method for the boundary stabilization of the wave equation. J Math Pures Appl. 1990;69:33–54.

    MathSciNet  MATH  Google Scholar 

  21. Lasiecka I. Global uniform decay rates for the solution to the wave equation with nonlinear boundary conditions. Appl Anal. 1992;47:191–212.

    Article  MathSciNet  MATH  Google Scholar 

  22. Lasiecka I. Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary. J Differ Equat. 1989;79:340–381.

    Article  MathSciNet  MATH  Google Scholar 

  23. Lasiecka I, Tataru D. Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping. Differ Integr Equ. 1993;8:507–533.

    MathSciNet  Google Scholar 

  24. Lasiecka I, Toundykov D. Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms. Nonlinear Anal. 2006;64:1757–1797.

    Article  MathSciNet  MATH  Google Scholar 

  25. Lasiecka I, Toundykov D. Regularity of higher energies of wave equation with nonlinear localized damping and a nonlinear source. Nonlinear Anal. 2008;69:898–910.

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu K. Locally distributed control and damping for the conservative systems. SIAM J Control Optim. 1997;35:1574–1590.

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu W-J, Zuazua E. Decay rates for dissipative wave equations. Ricerche Mat. 1999;48:61–75.

    MathSciNet  MATH  Google Scholar 

  28. Martinez P. A new method to obtain decay rate estimates for dissipative systems. ESAIM Control Optim Calc Var. 1999;4:419–444.

    Article  MathSciNet  Google Scholar 

  29. Martinez P. A new method to obtain decay rate estimates for dissipative systems with localized damping. Rev Mat Complut. 1999;12(1):251–283.

    MathSciNet  MATH  Google Scholar 

  30. Messaoudi SA. General decay of solutions of a viscoelastic equation. J Math Anal Appl. 2008;341:1457–1467.

    Article  MathSciNet  Google Scholar 

  31. Messaoudi SA. General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Anal. 2008;69:2589–2598.

    Article  MathSciNet  MATH  Google Scholar 

  32. Messaoudi SA, Mustafa MI. On convexity for energy decay rates of a viscoelastic equation with boundary feedback. Nonlinear Anal TMA. 2010;72:3602–3611.

    Article  MathSciNet  MATH  Google Scholar 

  33. Munoz Rivera JE. Asymptotic behavior in linear viscoelasticity. Quart Appl Math. 1994;52(4):628–648.

    MathSciNet  Google Scholar 

  34. Munoz Rivera JE, Oquendo HP. Exponential stability to a contact problem of partially viscoelastic materials. J Elasticity. 2001;63(2):87–111.

    Article  MathSciNet  MATH  Google Scholar 

  35. Munoz Rivera JE, Salvatierra AP. Asymptotic behavior of the energy in partially viscoelastic materials. Quart Appl Math. 2001;59(3):557–578.

    MathSciNet  MATH  Google Scholar 

  36. Munoz Rivera JE, Naso MG, Vegni FM. Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory. J Math Anal Appl. 2003;286(2):692–704.

    Article  MathSciNet  MATH  Google Scholar 

  37. Munoz Rivera JE, Naso MG. On the decay of the energy for systems with memory and indefinite dissipation. Asympt Anal. 2006;49(3–4):189–204.

    MathSciNet  MATH  Google Scholar 

  38. Nakao M. Decay of solutions of the wave equation with a local nonlinear dissipation. Math Ann. 1996;305: 403–417.

    Article  MathSciNet  MATH  Google Scholar 

  39. Zuazua E. Exponential decay for the semilinear wave equation with locally distributed damping. Comm Partial Differential Equations. 1990;15:205–235.

    Article  MathSciNet  MATH  Google Scholar 

  40. Zuazua E. Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J Control Optim. 1990;28:466–478.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author thanks KFUPM for its continuous support. This work has been funded by KFUPM under Project #IN121046.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O.Box 860, Dhahran, 31261, Saudi Arabia

    Muhammad I. Mustafa

Authors
  1. Muhammad I. Mustafa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Muhammad I. Mustafa.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mustafa, M.I. Uniform Decay Rates for Viscoelastic Dissipative Systems. J Dyn Control Syst 22, 101–116 (2016). https://doi.org/10.1007/s10883-014-9256-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-014-9256-1

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation