Skip to main content
SpringerLink
Log in

General decay of solutions for a weak viscoelastic equation with acoustic boundary conditions

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this paper, we consider the weak viscoelastic equation

$$u_{tt} - \Delta u + \alpha(t) \int\limits_{0}^{t} g(t-s)\Delta u(s)\, {\rm d}s=0$$

with a homogeneous Dirichlet condition on a portion of the boundary and acoustic boundary conditions on the rest of the boundary. We establish a general decay result, which depends on the behavior of both α and g, by using the perturbed energy functional technique. This is an extension and improvement of the previous result from Park and Park (Nonlinear Anal 74(3):993–998, 2011) (i.e., the similar problem with \({\alpha(t) \equiv 1}\)) to the time-dependent viscoelastic case.

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. Beale J.T.: Spectral properties of an acoustic boundary condition. Indiana Univ. Math. J. 25(9), 895–917 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  2. Beale J.T.: Acoustic scattering from locally reacting surfaces. Indiana Univ. Math. J. 26(2), 199–222 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  3. Beale J.T., Rosencrans S.I.: Acoustic boundary conditions. Bull. Am. Math. Soc. 80, 1276–1278 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berrimi S., Messaoudi S.A.: Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping. Electron. J. Differ. Equ. 2004(88), 10 (2004)

    MathSciNet  Google Scholar 

  5. Berrimi S., Messaoudi S.A.: Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal. 64(10), 2314–2331 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cavalcanti M.M., Domingos Cavalcanti V.N., Ferreira J.: Existence and uniform decay for a non-linear viscoelastic equation with strong damping. Math. Methods Appl. Sci. 24(14), 1043–1053 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cavalcanti M.M., Domingos Cavalcanti V.N., Martinez P.: General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal. 68(1), 177–193 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cavalcanti, M.M., Oquendo, H.P.: Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim. 42(4), 1310–1324 (2003) (electronic)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Dafermos C.M.: An abstract volterra equation with applications to linear viscoelasticity. J. Differ. Equ. 7, 554–569 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  11. Frota C.L., Goldstein J.A.: Some nonlinear wave equations with acoustic boundary conditions. J. Differ. Equ. 164(1), 92–109 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Frota, C.L., Larkin, N.A.: Uniform stabilization for a hyperbolic equation with acoustic boundary conditions in simple connected domains, in Contributions to nonlinear analysis, 297–312, Progr. Nonlinear Differential Equations Appl., 66 Birkhäuser, Basel, (2006)

  13. Liu W.J., Yu J.: On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms. Nonlinear Anal. 74(6), 2175–2190 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Messaoudi S.A.: General decay of solutions of a viscoelastic equation. J. Math. Anal. Appl. 341(2), 1457–1467 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Messaoudi S.A.: General decay of solutions of a weak viscoelastic equation. Arab. J. Sci. Eng. 36(8), 1569–1579 (2011)

    Article  MathSciNet  Google Scholar 

  16. Muñoz Rivera J.E.: Asymptotic behaviour in linear viscoelasticity. Quart. Appl. Math. 52(4), 628–648 (1994)

    MathSciNet  Google Scholar 

  17. Muñoz Rivera J.E., Lapa E.C.: Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomially decaying kernels. Comm. Math. Phys. 177(3), 583–602 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Muñoz Rivera J.E., Lapa E.C., Barreto R.: Decay rates for viscoelastic plates with memory. J. Elasticity 44(1), 61–87 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  19. Muñoz Rivera J.E., Qin Y.: Polynomial decay for the energy with an acoustic boundary condition. Appl. Math. Lett. 16(2), 249–256 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Park, J.Y., Ha, T.G.: Well-posedness and uniform decay rates for the Klein-Gordon equation with damping term and acoustic boundary conditions, J. Math. Phys. 50(1), 013506, (2009) 18 pp

    Google Scholar 

  21. Park J.Y., Park S.H.: Decay rate estimates for wave equations of memory type with acoustic boundary conditions. Nonlinear Anal. 74(3), 993–998 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  22. Tahamtani F., Shahrouzi M.: Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source term. Boundary Value Probl. 2012(50), 1–15 (2012)

    MathSciNet  Google Scholar 

  23. Vicente A., Frota C.L.: Nonlinear wave equation with weak dissipative term in domains with non-locally reacting boundary. Wave Motion 50(2), 162–169 (2013)

    Article  MathSciNet  Google Scholar 

  24. Wu S.-T.: General decay of energy for a viscoelastic equation with damping and source terms. Taiwanese J. Math. 16(1), 113–128 (2012)

    MATH  MathSciNet  Google Scholar 

  25. Wu B., Yu J., Wang Z.: Uniqueness and stability of an inverse kernel problem for type III thermoelasticity. J. Math. Anal. Appl. 402(1), 242–254 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  26. Zhang Z. et al.: A note on decay properties for the solutions of a class of partial differential equation with memory. J. Appl. Math. Comput. 37(1–2), 85–102 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Wenjun Liu & Yun Sun

Authors
  1. Wenjun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yun Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenjun Liu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, W., Sun, Y. General decay of solutions for a weak viscoelastic equation with acoustic boundary conditions. Z. Angew. Math. Phys. 65, 125–134 (2014). https://doi.org/10.1007/s00033-013-0328-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-013-0328-y

Mathematics Subject Classification (2000)

Keywords

Search

Navigation