Exponential stabilization of a viscoelastic wave equation with dynamic boundary conditions



In this paper, we study the stabilization of a semi-linear viscoelastic wave equation subject to semi-linear and dynamical boundary conditions. The kernels used are of strongly positive definite type. We prove that internal and boundary memory damping are strong enough, via transmission process (\({u|_{\Gamma} = v}\)), to stabilize the whole system.


Wave equation Viscoelastic Dynamical boundary conditions Global existence Asymptotic stability Strongly positive definite kernels 

Mathematics Subject Classification

93D20 35L15 35L70 45M10 65M60 45N05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aassila M., Cavalcanti M.M., Soriano J.A.: Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a starshaped domain. SIAM J. Control Optim. 38(5), 1581–1602 (2000)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Alabau-Boussouira F., Cannarsa P., Sforza D.: Decay estimates for second order evolution equations with memory. J. Funct. Anal. 254, 1342–1372 (2008)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Cannarsa P., Sforza D.: integro-differential equations of hyperbolic type with positive definite kernels. J. Differ. Equ. 250, 4289–4335 (2011)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Cannarsa P., Sforza D.: A stability result for a class of nonlinear integrodifferential equations with L 1 kernels. Appl. Math. (Warsaw) 35, 395–430 (2008)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Cavalcanti M.M., Domingos Cavalcanti V.N., Ferreira J.: Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math. Methods Appl. Sci. 24, 1043–1053 (2001)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Cavalcanti M.M., Domingos Cavalcanti V.N., Prates Filho J.S., Soriano J.A.: Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differ. Integr. Equ. 14, 85–116 (2001)MATHMathSciNetGoogle Scholar
  7. 7.
    Cavalcanti M.M., Domingos Cavalcanti V.N., Soriano J.A.: Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron. J. Differ. Equ. 44, 1–14 (2002)MathSciNetGoogle Scholar
  8. 8.
    Cavalcanti M.M., Khemmoudj A., Medjden M.: Uniform stabilization of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary condition. J. Math. Anal. Appl 328(2), 900–930 (2007)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Cavalcanti M.M., Oquendo H.P.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42(4), 1310–1324 (2003)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Dafermos C.M.: Asymptotic stability in viscoelasticity. Arch. Rational. Mech. Anal. 37, 297–308 (1970)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Dafermos C.M.: On abstract Volterra equation with applications to linear viscoeladticity. J. Differ. Equ. 7, 554–569 (1970)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Fabrizio M., Giorgi C., Pata V.: A new approach to equations with memory. Arch. Rational. Mech. Anal. 198, 189–232 (2010)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Gripenberg, G., Londen, S.O., Staffans, O.J.: Volterra Integral and Functional Equations. Encyclopedia Math. Appl., vol. 34. Cambridge Univ. Press, Cambridge (1990)Google Scholar
  14. 14.
    Hrusa W.J., Nohel J.A.: The Cauchy problem in one-dimensional nonlinear viscoelasticity. J. Differ. Equ. 59, 388–412 (1985)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Kawashima S.: Global solutions to the equation of viscoelasticity with fading memory. J. Differ. Equ. 101, 388–420 (1993)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Khemmoudj A., Medjden M.: Exponential decay for the semilinear damped Cauchy-Ventcel problem. Bol. Soc. Parana. Mat. 22(2), 97–116 (2004)MATHMathSciNetGoogle Scholar
  17. 17.
    Lasiecka I., Triggiani R., Yao P.F.: Inverse/observability estimates for second-ordre hyperbolic equations with variable coefficients. J. Math. Anal. Appl. 235(1), 13–57 (1999)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Nicaise, S., Laoubi, K.: Polynomial stabilization of the wave equation with Ventcel’s boundary conditions. Math. Nachr. 283, 10 (2010)Google Scholar
  19. 19.
    Prüss, J.: Evolutionary Integral Equations and Applications. Monogr. Math., vol. 87. Birkhäuser Verlag, Basel (1993)Google Scholar
  20. 20.
    Staffans O.J.: Positive definite measures with applications to a Volterra equation. Trans. Am. Math. Soc. 218, 219–237 (1976)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Staffans O.J.: On a nonlinear hyperbolic Volterra equation. SIAM J. Math. Anal. 11, 793–812 (1980)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Zuazua E.: Exponential decay for the semilinear wave equation with locally distributed damping. Commun. Partial Differ. Equ. 15(2), 205–235 (1990)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of Sciences and Technology Houari BoumedieneBab EzzouarAlgeria

Personalised recommendations