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Existence and energy decay of a Bresse system with thermoelasticity of type III

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Abstract

In this paper, we investigate a one-dimensional thermoelastic Bresse system, where the heat conduction is given by Green and Naghdi theories. Under some assumptions on the memory kernel and a new introduced stability number, we prove that the unique damping given by the memory term is sufficiently strong to stabilize the system exponentially. In fact, we establish a general decay result from which the exponential and polynomial decays are only special cases.

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References

  1. Afilal, M., Soufyane, A., Radid, A.: New decay rates for Cauchy problem of the Bresse system in thermoelasticty type III. Appl. Anal. 100(14), 2911–2926 (2021)

    Article  MathSciNet  Google Scholar 

  2. Boussouira, F.A., Muñoz Rivera, J.E., Almeida Júnior, D.S.: Stability to weak dissipative Bresse system. J. Math. Anal. Appl. 374(2), 481–498 (2011)

    Article  MathSciNet  Google Scholar 

  3. Bresse, J.A.C.: Cours de mecanique appliquee par M. Bresse: Résistance des matériaux et stabilité des constructions. Mallet-Bachelier, Paris (1859)

    Google Scholar 

  4. Cattaneo, C.: Sulla conduzione del calore. Atti Sem. Mat. Fis. Univ. Modena. 3, 83–101 (1948)

    MathSciNet  MATH  Google Scholar 

  5. Copetti, M.I.M., El Arwadi, T., Fernández, J.R., Naso, M.G., Youssef, W.: Analysis of a contact problem for a viscoelastic Bresse system. ESAIM Math. Model. Numer. Anal. 55(3), 887–911 (2021)

    Article  MathSciNet  Google Scholar 

  6. De Lima, P.R., Fernández Sare, H.D.: Stability of thermoelastic Bresse systems. Z. Angew. Math. Phys. 70, 1–33 (2018)

    MathSciNet  MATH  Google Scholar 

  7. Dell’Oro, F.: Asymptotic stability of thermoelastic systems of Bresse type. J. Differ. Equ. 258(11), 3902–3927 (2015)

  8. Djebabla, A., Tatar, N.: Exponential stabilization of the Timoshenko system by a thermo-viscoelastic damping. J. Dyn. Control. Syst. 16(2), 189–210 (2010)

    Article  MathSciNet  Google Scholar 

  9. Djebabla, A., Tatar, N.: Exponential stabilization of the Timoshenko system by a thermal effect with an oscillating kernel. Math. Comput. Model. 54(1–2), 301–314 (2011)

    Article  MathSciNet  Google Scholar 

  10. Djellali, F., Labidi, S., Taallah, F.: General decay for a viscoelastic-type Timoshenko system with thermoelasticity of type III. Appl. Anal. (2021). https://doi.org/10.1080/00036811.2021.1967329

    Article  MATH  Google Scholar 

  11. El Arwadi, T., Youssef, W.: On the stabilization of the Bresse beam with Kelvin–Voigt damping. Appl. Math. Optim. 83(3), 1831–1857 (2021)

    Article  MathSciNet  Google Scholar 

  12. Fatori, L.H., Muñoz Rivera, J.E.: Rates of decay to weak thermoelastic Bresse system. IMA J. Appl. Math. 75(6), 881–904 (2010)

    Article  MathSciNet  Google Scholar 

  13. Green, A.E., Naghdi, P.M.: A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. A 432(1885), 171–194 (1991)

    Article  MathSciNet  Google Scholar 

  14. Green, A.E., Naghdi, P.M.: On undamped heat waves in an elastic solid. J. Therm. Stresses 15(2), 253–264 (1992)

    Article  MathSciNet  Google Scholar 

  15. Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 31(3), 189–208 (1993)

    Article  MathSciNet  Google Scholar 

  16. Guesmia, A., Kirane, M.: Uniform and weak stability of Bresse system with two infinite memories. Z. Angew. Math. Phys. 67(5), 1–39 (2016)

    Article  MathSciNet  Google Scholar 

  17. Guesmia, A.: The effect of the heat conduction of types I and III on the decay rate of the Bresse system via the longitudinal displacement. Arab. J. Math. 8(1), 15–41 (2019)

    Article  MathSciNet  Google Scholar 

  18. Gurtin, M.E., Pipkin, A.C.: A general theory of heat conduction with finite wave speeds. Arch. Ration. Mech. Anal. 31(2), 113–126 (1968)

    Article  MathSciNet  Google Scholar 

  19. Houasni, M., Zitouni, S., Amiar, R.: Exponential decay for a thermo-viscoelastic Bresse system with second sound and delay terms. Int. J. Maps Math. 3(1), 39–56 (2020)

    MathSciNet  Google Scholar 

  20. Keddi, A.A., Apalara, T.A., Messaoudi, S.A.: Exponential and polynomial decay in a thermoelastic-Bresse system with second sound. Appl. Math. Optim. 77(2), 315–341 (2018)

    Article  MathSciNet  Google Scholar 

  21. Liu, Z., Rao, B.: Energy decay rate of the thermoelastic Bresse system. Z. Angew. Math. Phys. 60(1), 54–69 (2009)

    Article  MathSciNet  Google Scholar 

  22. Messaoudi, S.A., Hassan, J.H.: New general decay results in a finite-memory Bresse system. Commun. Pure Appl. Anal. 18(4), 1637–1662 (2019)

    Article  MathSciNet  Google Scholar 

  23. Santos, M.L., Soufyane, A., Almeida Junior, D.S.: Asymptotic behavior to Bresse system with past history. Q. Appl. Math. 73(1), 23–54 (2015)

    Article  MathSciNet  Google Scholar 

  24. Santos, M.L.: Bresse system in thermoelasticity of type III acting on shear force. J. Elast. 125(2), 185–216 (2016)

    Article  MathSciNet  Google Scholar 

  25. Soriano, J.A., Charles, W., Schulz, R.: Asymptotic stability for Bresse systems. J. Math. Anal. Appl. 412(1), 369–380 (2014)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematical Modeling and Numerical Simulation Laboratory, Badji-Mokhtar University, P.O. Box 12, 23000, Annaba, Algeria

    F. Djellali & F. Taallah

  2. Laboratory of Applied Mathematics, Badji-Mokhtar University, P.O. Box 12, 23000, Annaba, Algeria

    S. Labidi

Authors
  1. F. Djellali
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  2. S. Labidi
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  3. F. Taallah
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Djellali, F., Labidi, S. & Taallah, F. Existence and energy decay of a Bresse system with thermoelasticity of type III. Z. Angew. Math. Phys. 73, 3 (2022). https://doi.org/10.1007/s00033-021-01641-4

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  • DOI: https://doi.org/10.1007/s00033-021-01641-4

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