Skip to main content
SpringerLink
Log in

Uniform stability of a thermodiffusion Timoshenko beam

  • Original Paper
  • Published:
Partial Differential Equations and Applications Aims and scope Submit manuscript

Abstract

The main object of the present work is the study of a new Timoshenko beam model with thermal and mass diffusion effects where the coupling is acting on the shear force. We prove the well-posedness of the system using the semigroup theory. Furthermore, we establish that the system is exponentially stable if and only if the wave speeds of the system are equal. When the speeds of the mechanical waves are different, we show a lack of exponential stability. Additionally, in the case of different wave speeds, we show that the solution decays polynomially.

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

Data Availability

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

References

  1. Almeida Júnior, D.S., Santos, M.L., Muñoz Rivera, J.E.: Stability to 1-D thermoelastic Timoshenko beam acting on shear force. Z. Angew. Math. Phys. 65(6), 1233–1249 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alves, M.O., Caixeta, A.H., Jorge Silva, M.A., Rodrigues, J.H., Almeida Júnior, D.S.: On a Timoshenko system with thermal coupling on both the bending moment and the shear force. J. Evol. Equ. 20, 295–320 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alves, M.S., Jorge Silva, M.A., Ma, T.F., Muñoz Rivera, J.E.: Invariance of decay rate with respect to boundary conditions in thermoelastic Timoshenko systems. Z. Angew. Math. Phys. 67, 70 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ammar-Khodja, F., Benabdallah, A., Muñoz Rivera, J.E., Racke, R.: Energy decay for Timoshenko systems of memory type. J. Differ. Equ. 194(1), 82–115 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aouadi, M., Ciarletta, M., Tibullo, V.: Well-posedness and exponential stability in binary mixtures theory for thermoviscoelastic diffusion materials. J. Therm. Stress. 41, 1414–1431 (2018)

    Article  Google Scholar 

  6. Aouadi, M., Copetti, M.I.M.: A dynamic contact problem for a thermoelastic diffusion beam with the rotational inertia. Appl. Numer. Math. 126, 113–137 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Aouadi, M., Campo, M., Copetti, M.I.M., Fernández, J.R.: Existence, stability and numerical results for a Timoshenko beam with thermodiffusion effects. Z. Angew. Math. Phys. 70(4), 1–26 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Aouadi, M., Ramos, A., Castejón, A.: Stability conditions for thermodiffusion Timoshenko system with second sound. Z. Angew. Math. Phys. 72(4), 1–32 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  9. Apalara, T.A.: General stability of memory-type thermoelastic Timoshenko beam acting on shear force. Continuum Mech. Thermodyn. 30, 291–300 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Apalara, T.A., Raposo, C.A., Ige, A.: Thermoelastic Timoshenko system free of second spectrum. Appl. Math. Lett. 126, 107793 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  11. Djellali, F., Labidi, S., Taallah, F.: Exponential stability of thermoelastic Timoshenko system with Cattaneo’s law. Ann. Univ. Ferrara 67, 43–57 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Djellali, F., Labidi, S.: On the stability of a thermodiffusion Bresse system. J. Math. Phys. 63(8), 081505 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  13. Djellali, F., Labidi, S., Taallah, F.: General decay for a viscoelastic-type Timoshenko system with thermoelasticity of type III. Appl. Anal. 102, 902–920 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dridi, H., Feng, B., Zennir, K.: Stability of Timoshenko system coupled with thermal law of Gurtin–Pipkin affecting on shear force. Appl. Anal. 101, 5171–5192 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  15. Elhindi, M., El Arwadi, T.: Analysis of the thermoviscoelastic Timoshenko system with diffusion effect. Partial Differ. Equ. Appl. Math. 4, 100156 (2021)

    Article  Google Scholar 

  16. Elhindi, M., Zennir, K., Ouchenane, D., Choucha, A., El Arwadi, T.: Bresse-Timoshenko type systems with thermodiffusion effects: well-posedness, stability and numerical results. Rend. Circ. Mat. Palermo Ser. 2, 1–26 (2021)

    MATH  Google Scholar 

  17. Feng, B.: Exponential stabilization of a Timoshenko system with thermodiffusion effects. Z. Angew. Math. Phys. 72(4), 1–17 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Feng, B., Youssef, W., El Arwadi, T.: Polynomial and exponential decay rates of a laminated beam system with thermodiffusion effects. J. Math. Anal. Appl. 517(2), 126633 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gearhart, L.: Spectral theory for contraction semigroups on Hilbert space. Trans. Am. Math. Soc. 236, 385–394 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  20. Guesmia, A., Messaoudi, S.A., Soufyane, A.: Stabilization of a linear Timoshenko system with infinite history and applications to the Timoshenko-heat systems. Electron. J. Differ. Equ. 2012, 1–45 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Guesmia, A., Soufyane, A.: On the stability of Timoshenko-type systems with internal frictional dampings and discrete time delays. Appl. Anal. 96, 2075–2101 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Huang, F.: Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ. 1(1), 43–56 (1985)

    MathSciNet  MATH  Google Scholar 

  23. Liu, Z., Zheng, S.: Semigroups Associated with Dissipative Systems. Chapman Hall/CRC, Boca Raton (1999)

    MATH  Google Scholar 

  24. Messaoudi, S.A., Mustafa, M.I.: A general stability result in a memory-type Timoshenko system. Commun. Pure Appl. Anal. 12, 957–972 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Muñoz Rivera, J.E., Racke, R.: Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability. J. Math. Anal. Appl. 276(1), 248–278 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Muñoz Rivera, J.E., Fernández Sare, H.D.: Stability of Timoshenko systems with past history. J. Math. Anal. Appl. 339, 482–502 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    Book  MATH  Google Scholar 

  28. Prüss, J.: On the spectrum of \(C_{0}\)-semigroups. Trans. Am. Math. Soc. 284(2), 847–857 (1984)

    MATH  Google Scholar 

  29. Ramos, A.J.A., Aouadi, M., Almeida Júnior, D.S., Freitas, M.M., Araujo, M.L.: A new stabilization scenario for Timoshenko systems with thermo-diffusion effects in second spectrum perspective. Arch. Math. 116(2), 203–219 (2021)

  30. Soufyane, A.: Stabilisation de la poutre de Timoshenko. C. R. Acad. Sci. Ser. 1 Math 328, 731–734 (1999)

    MathSciNet  MATH  Google Scholar 

  31. Tian, X., Zhang, Q.: Stability of a Timoshenko system with local Kelvin-Voigt damping. Electron. Z. Angew. Math. Phys. 68, 1–15 (2017)

    MathSciNet  MATH  Google Scholar 

  32. Timoshenko, S.P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Lond. Edinb. Dubl. Philos. Mag. J. Sci. 41(245), 744–746 (1921)

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank very much both reviewers for their careful reading and valuable suggestions. The second author appreciates the continuous support of University of Hafr Al Batin.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, Badji-Mokhtar University, PO Box 12, 23000, Annaba, Algeria

    Fayssal Djellali & Mohamed Zitouni

  2. Department of Mathematics, University of Hafr Al-Batin (UHB), Hafr Al-Batin, 31991, Saudi Arabia

    Tijani A. Apalara

Authors
  1. Fayssal Djellali
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tijani A. Apalara
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohamed Zitouni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fayssal Djellali.

Ethics declarations

Conflict of interest

The authors have no competing interests to declare that are relevant to the content of this article.

Additional information

This article is part of the section “Theory of PDEs” edited by Eduardo Teixeira.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Djellali, F., Apalara, T.A. & Zitouni, M. Uniform stability of a thermodiffusion Timoshenko beam. Partial Differ. Equ. Appl. 4, 22 (2023). https://doi.org/10.1007/s42985-023-00243-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s42985-023-00243-1

Keywords

Mathematics Subject Classification

Search

Navigation