Skip to main content
SpringerLink
Log in

Stabilization of non-homogeneous viscoelastic waves coupled with heat conduction due to Cattaneo-Vernotte

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

We study the stabilization of non-homogeneous viscoelastic waves for the vibrations of a flexible structure with thermodiffusion effect and a distributed forcing term as input disturbance. The coupled heat conduction is governed by Cattaneo-Vernotte’s law. Using the semigroup theory, we prove the existence and the uniqueness of the solution. Under construction of a suitable Lyapunov functional, it is shown that the amplitude of such vibrations is bounded for admissible bounded input disturbances. An estimate of the total energy of the system over a time interval is obtained directly with a tolerance level of the disturbance. Finally, in the absence of input disturbance, the uniform exponential decay of solution with an explicit form of energy decay estimate is achieved directly.

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. Alves, M.S., Gamboa, P., Gorain, G.C., Rambaud, A., Vera, O.: Asymptotic behavior of a flexible structure with Cattaneo type of thermal effect. Indag. Math. 27(3), 821–834 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Auriault, J.L.: Cattaneo-Vernotte equation versus Fourier thermoelastic hyperbolic heat equation. Int. J. Eng. Sci. 101(1), 45–49 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chandrasekharaiah, D.S.: Hyperbolic thermoelasticity: A review of recent literature. Appl. Mech. Rev. 51(12), 705–728 (1998)

    Article  Google Scholar 

  4. Chen, G.: Energy decay estimate and exact boundary-value controllability for the wave equation in a bounded domain. J. Math. Pures Appl. 58, 249–273 (1979)

    MathSciNet  MATH  Google Scholar 

  5. Chen, G.: A note on the boundary stabilization of the wave equation. SIAM J. Control Optim. 19(1), 106–113 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  6. Choi, J.H., Yoon, S.H., Park, S.G., Choi, S.H.: Analytical solution of the Cattaneo Vernotte equation (non-Fourier heat conduction). J. Korean Soc. Marine Eng. 40(5), 389–396 (2016)

    Google Scholar 

  7. Christensen, R.M.: Theory of viscoelasticity. Academic Press, New York (1971)

    Google Scholar 

  8. Day, W.A.: Heat Conduction Within Linear Thermoelasticity. Springer, New york (1985)

    Book  MATH  Google Scholar 

  9. Gorain, G.C.: Exponential energy decay estimate for the solutions of internally damped wave equation in a bounded domain. J. Math. Anal. Appl. 216(2), 510–520 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gorain, G.C.: Exponential energy decay estimate for the solutions of n-dimensional Kirchhoff type wave equation. Appl. Math. Comput. 177(1), 235–242 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Gorain, G.C.: Boundary stabilization of nonlinear vibrations of a flexible structure in a bounded domain in \({{\mathbf{R}}}^{n}\). J. Math. Anal. Appl. 319(2), 635–650 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gorain, G.C.: Stabilization of a quasi-linear vibrations of an inhomogeneous beam. IEEE Trans. Automat. Control. 52(9), 1690–1695 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gorain, G.C.: Exponential stabilization of longitudinal vibrations of an inhomogeneous beam. J. Math. Sci. 198(3), 245–251 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ho, L.F.: Exact controllability of the one-dimensional wave equation with locally distributed control. SIAM J. Control Optim. 28(3), 733–748 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kim, J.U.: On the energy decay of a linear thermoelastic bar and plate. SIAM J. Math. Anal. 23(4), 889–899 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Komornik, V., Zuazua, E.: A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. 69(1), 33–54 (1990)

    MathSciNet  MATH  Google Scholar 

  17. Komornik, V.: Rapid boundary stabilization of wave equations. SIAM J. Control Optim. 29(1), 197–208 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Komornik, V.: Exact Controllability and Stabilization. The Multiplier Method, Wiley, Paris (1994)

    MATH  Google Scholar 

  19. Lagnese, J.: Note on boundary stabilization of wave equations. SIAM J. Control Optim. 26(5), 1250–1256 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lasiecka, I., Lions, J.L., Triggiani, R.: Non homogeneous boundary value problems for second order hyperbolic operators. J. Math. Pures Appl. 65(1), 149–192 (1986)

    MathSciNet  MATH  Google Scholar 

  21. Lions, J.L.: Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev. 30(1), 1–68 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  22. Marzocchi, A., Rivera, J.E.M., Naso, M.G.: Asymptotic behaviour and exponential stability for a transmission problem in thermoelasticity. Math. Methods Appl. Sci. 25(11), 955–980 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Messaoudi, S.A., Said-Houari, B.: Exponential stability in one-dimensional non-linear thermoelasticity with second sound. Math. Methods Appl. Sci. 28(2), 205–232 (2005)

    MathSciNet  MATH  Google Scholar 

  24. Misra, S., Alves, M., Gorain, G.C., Vera, O.: Stability of the vibrations of an inhomogeneous flexible structure with thermal effect. Int. J. Dyn. Control. 3(4), 354–362 (2015)

    Article  MathSciNet  Google Scholar 

  25. Mitrinović, D.S., Pec̆arić, J.E., Fink, A.M.: Inequalities Involving Functions and Their Integrals and Derivatives. Dordrecht, The Netherlands: Kluwer, (1991)

  26. Mochnacki, B., Paruch, M.: Cattaneo-Vernotte equation. Identification of relaxation time using evolutionary algorithms. J. Appl. Math. and Comp. Mech. 12(4), 97–102 (2013)

    Article  MATH  Google Scholar 

  27. Nandi, P.K., Gorain, G.C., Kar, S.: A note on stability of longitudinal vibrations of an inhomogeneous beam. Appl. Math. 3(1), 19–23 (2012)

    Article  MathSciNet  Google Scholar 

  28. Pazy, A.: Semigroup of Linear Operators and Applications to Partial Differential Equations. Springer, New york (1983)

    Book  MATH  Google Scholar 

  29. Prüss, J.: On the spectrum of \(C_0\)-semigroups. Trans. Amer. Math. Soc. 284(2), 847–857 (1984)

    MathSciNet  MATH  Google Scholar 

  30. Racke, R.: Thermoelasticity with second sound–exponential stability in linear and non-linear 1-d. Quart. Appl. Math. 25(5), 409–441 (2002)

    MathSciNet  MATH  Google Scholar 

  31. Racke, R.: Asymptotic behavior of solutions in linear 2-d or 3-d thermoelasticity with second sound. Quart. Appl. Math. 61(2), 315–328 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  32. Raposo, C.A., Santos, M.J.: Exponential stability for a transmission problem with locally indirect stabilization. Appl. Math. Inf. Sci. 13(1), 1–6 (2019)

    Article  MathSciNet  Google Scholar 

  33. Raposo, C.A., Apalara, T.A., Ribeiro, J.O.: Analyticity to transmission problem with delay in porous-elasticity. J. Math. Anal. Appl. 466(1), 819–834 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shahruz, S.M.: Bounded-input bounded-output stability of a damped nonlinear string. Trans. Auto. Control 41(8), 1179–1182 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The third author is partially financed by project Fondecyt 1191137.

Author information

Authors and Affiliations

  1. Department of Mathematics, J. K. College, Purulia, West Bengal, India

    Ganesh C. Gorain

  2. Department of Mathematics, Federal University of Bahia, Salvador, BA, Brazil

    Carlos A. Raposo

  3. Department of Mathematics, University of Tarapaca, Arica, Chile

    Octavio Vera

Authors
  1. Ganesh C. Gorain
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carlos A. Raposo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Octavio Vera
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carlos A. Raposo.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gorain, G.C., Raposo, C.A. & Vera, O. Stabilization of non-homogeneous viscoelastic waves coupled with heat conduction due to Cattaneo-Vernotte. Ricerche mat 72, 359–377 (2023). https://doi.org/10.1007/s11587-022-00722-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-022-00722-4

Keywords

Mathematics Subject Classification

Search

Navigation