Skip to main content
SpringerLink
Log in

Gevrey Class for Locally Three-Phase-Lag Thermoelastic Beam System

  • Published:
Applied Mathematics & Optimization Submit manuscript

Abstract

In this article we study the behavior of the solutions for the three-phase-lag heat equation with localized dissipation on an Euler–Bernoulli beam model. We show that semigroup S(t) associated with the problem is of Gevrey class 5 for \(t>0\). If the coefficients satisfy \(\tau _\alpha > k^{*}\tau _q\), the solutions are always exponentially stable.

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

Fig. 1

Similar content being viewed by others

References

  1. Cattaneo, C.: A form of heat equation which eliminates the paradox of instantaneous propagation. C. R. Acad. Sci. Paris 247, 431–433 (1958)

    MathSciNet  Google Scholar 

  2. Tzou, D.Y.: A unified approach for heat conduction from macro to micro-scales. ASME J. Heat Transf. 117, 8–16 (1995)

    Article  Google Scholar 

  3. Choudhuri, S.K.R.: On a thermoelastic three-phase-lag model. J. Therm. Stress. 30, 231–238 (2007)

    Article  Google Scholar 

  4. Green, A.E., Naghdi, P.M.: A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. A 432, 171–194 (1991)

    Article  MathSciNet  Google Scholar 

  5. Green, A.E., Naghdi, P.M.: On undamped heat waves in an elastic solid. J. Therm. Stress. 15, 253–264 (1992)

    Article  MathSciNet  Google Scholar 

  6. Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–208 (1993)

    Article  MathSciNet  Google Scholar 

  7. Green, A.E., Naghdi, P.M.: A unified procedure for construction of theories of deformable media. I. Classical continuum physics, II. Generalized continua, III. Mixtures of interacting continua. Proc. R. Soc. Lond. A 448, 335–356, 357–377, 379–388 (1995)

  8. Dreher, M., Quintanilla, R., Racke, R.: Ill-posed problems in thermomechanics. Appl. Math. Lett. 22, 1374–1379 (2009)

    Article  MathSciNet  Google Scholar 

  9. Quintanilla, R., Racke, R.: A note on stability in three-phase-lag heat conduction. Int. J. Heat Mass Transf. 51, 24–29 (2008)

    Article  Google Scholar 

  10. Borgmeyer, K.: Dual and three-phase-lag-modelle: Zeitliche Asymptotik von Lösungen. Dissertation (Ph.D. thesis), University of Konstanz (2016)

  11. Quintanilla, R.: Spatial behavior of solutions of the three-phase-lag heat equation. Appl. Math. Comput. 213, 153–162 (2009)

    MathSciNet  Google Scholar 

  12. Borgmeyer, K., Quintanilla, R., Racke, R.: Phase-lag heat conduction: decay rates for limit problems and well-posedness. J. Evol. Equ. 14, 863–884 (2014)

    Article  MathSciNet  Google Scholar 

  13. Liu, Z., Quintanilla, R., Wang, Y.: On the regularity and stability of three-phase-lag thermoelastic plate. Appl. Anal. 101, 5376–5385 (2022)

    Article  MathSciNet  Google Scholar 

  14. Sozzo, B.T.S., Rivera, J.E.M.: Gevrey class for locally thermoelastic beam equations. Z. Angew. Math. Phys. 73, 153 (2022)

    Article  MathSciNet  Google Scholar 

  15. Ávalos, G Gómez., Rivera, J Muñoz., Liu, Z.: Gevrey class of locally dissipative Euler-Bernoulli beam equation. SIAM J. Control Optim. 59, 2174–2194 (2021)

    Article  MathSciNet  Google Scholar 

  16. Ávalos, G.G., Rivera, J.M., Ochoa, E.O.: Gevrey semigroup of the type III localized thermoelastic model. J. Differ. Equ. 347, 282–309 (2023)

    Article  MathSciNet  Google Scholar 

  17. Choudhuri, S.K.R.: On a thermoelastic three-phase-lag model. J. Therm. Stress. 30, 231–238 (2007)

    Article  Google Scholar 

  18. Taylor, S.: Chapter “Gevrey semigroups”. Ph.D. thesis, School of Mathematics, University of Minnesota (1989)

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

    Book  Google Scholar 

  20. Crandall, M.G., Pazy, A.: On the differentiability of weak solutions of a differential equation in Banach space. J. Math. Mech. 10, 1007–1016 (1969)

    MathSciNet  Google Scholar 

  21. Davies, E.B.: One Parameter Semigroups. London Mathematical Society Monographs, vol. 15, pp. 115–116. Academic Press, New York (1980)

    Google Scholar 

  22. Engel, K.-J., Nagel, R.: One Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)

    Google Scholar 

  23. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    Google Scholar 

Download references

Funding

Jaime Muñoz Rivera was supported by CNPq project 307947/2022-0 and Fondecyt project 1230914. Elena Ochoa was supported by CONICYT-PFCHA/doctorado nacional/2020-21200268. Ramón Quintanilla was supported by the project PID2019-105118GB-I00, funded by the Spanish Ministry of Science, Innovation and Universities and FEDER “A way to make Europe”.

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Facultad de Ciencias, Universidad del Bío-Bío, Avenida Collao 1202, Concepción, Chile

    Jaime Muñoz Rivera

  2. Departamento de Matemáticas, Facultad de Ciencias Exactas, Universidad Andres Bello, Sede Concepción, Autopista Concepción-Talcahuano 7100, Talcahuano, Chile

    Elena Ochoa Ochoa

  3. Departamento de Matemáticas, Universidad Politécnica de Catalunya, Colom 11, 08222, Terrassa, Spain

    Ramón Quintanilla

Authors
  1. Jaime Muñoz Rivera
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elena Ochoa Ochoa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ramón Quintanilla
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elena Ochoa Ochoa.

Ethics declarations

Conflict of interest

This work does not have any conflicts 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

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

Rivera, J.M., Ochoa, E.O. & Quintanilla, R. Gevrey Class for Locally Three-Phase-Lag Thermoelastic Beam System. Appl Math Optim 89, 51 (2024). https://doi.org/10.1007/s00245-024-10125-6

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00245-024-10125-6

Keywords

Search

Navigation