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Stabilization of a Microtemperature Porous-Elastic System with Distributed Delay-Time

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Abstract

In this paper, we consider a one-dimensional porous-elastic system with microtemperature and distributed delay term acting on the porous equation. Under suitable assumptions on the weight of distributed delay, we establish the well-posedness of the system by using semigroup theory and we prove that the unique dissipation due to the microtemperature is strong enough to exponentially stabilize the system when the speeds of wave propagation are equal.

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References

  1. Apalara, T.: Exponential decay in one-dimensional porous dissipation elasticity. Q J Mech. Appl. Math. 70(4), 553–555 (2017). https://doi.org/10.1093/qjmam/hbx027

    Article  MathSciNet  MATH  Google Scholar 

  2. Apalara, T.: On the stability of porous-elastic system with microtemparatures. J. Therm. Stress 42(2), 265–278 (2019). https://doi.org/10.1080/01495739.2018.1486688

    Article  Google Scholar 

  3. Filho, E.B., Santos, M.L.: On porous-elastic system with a time-varying delay term in the internal feedbacks, Z Angew Math Mech. e201800247. (2020). https://doi.org/10.1002/zamm.201800247

  4. Casas, P.S., Quintanilla, R.: Exponential decay in one-dimensional porous-thermo-elasticity. Mech. Res. Commun. 32, 652–658 (2005)

    Article  MathSciNet  Google Scholar 

  5. Casas, P.S., Quintanilla, R.: Exponential stability in thermoelasticity with microtemperatures. Int. J. Eng. Sci. 43, 33–47 (2005)

    Article  MathSciNet  Google Scholar 

  6. Chirita, S., Ciarletta, M., D’Apice, C.: On the theory of thermoelasticity with microtemperatures. J. Math. Anal. Appl. 397(1), 349–361 (2013). https://doi.org/10.1016/j.jmaa.2012.07.06

    Article  MathSciNet  MATH  Google Scholar 

  7. Cowin, S.C., Nunziato, J.W.: Linear elastic materials with voids. J Elast 13(2), 125–147 (1983). https://doi.org/10.1007/BF00041230

    Article  MATH  Google Scholar 

  8. Cowin, S.C.: The viscoelastic behavior of linear elastic materials with voids. J Elast. 15(2), 185–191 (1985). https://doi.org/10.1007/BF00041992

    Article  MathSciNet  MATH  Google Scholar 

  9. Goodman, M.A., Cowin, S.C.: A continuum theory for granular materials. Arch. Ration Mech. Anal. 44(4), 249–266 (1972)

    Article  MathSciNet  Google Scholar 

  10. Han, Z.J., Xu, G.Q.: Exponential decay result in non-uniform porous-thermo-elasticity model of Lord-Shulman type. Disc. Cont. Dyn. Sys. B 17, 57–77 (2012). https://doi.org/10.3934/dcdsb.2012.17.57

    Article  MATH  Google Scholar 

  11. Hao, J., Wang, P.: Exponential decay of solution to the viscoelastic porous-thermoelastic system of type III with boundary time-varying delay. Math. Methods Appl. Sci. 39(13), 3659–3668 (2016)

    Article  MathSciNet  Google Scholar 

  12. Khochemane, H.E., Bouzettouta, L., Guerouah, A.: Exponential decay and well-posedness for a one-dimensional porous-elastic system with distributed delay, Applicable Analysis, https://doi.org/10.1080/00036811.2019.1703958

  13. Liu, W., Chen, M.: Well-posedness and exponential decay for a porous thermoelastic system with second sound and a time-varying delay term in the internal feedback. Contin. Mech. Thermodyn. 29, 731–746 (2017). https://doi.org/10.1007/s00161-017-0556-z

    Article  MathSciNet  MATH  Google Scholar 

  14. Lesan, D.I., Quintanilla, R.: A theory of porous thermoviscoelastic mixtures. J. Thermal Stresses 30(7), 693–714 (2007). https://doi.org/10.1080/01495730701212880

    Article  MathSciNet  Google Scholar 

  15. Lesan, D.I.: A theory of thermoelastic materials with voids. Acta Mech. 60(1–2), 67–89 (1986). https://doi.org/10.1007/BF01302942

    Article  Google Scholar 

  16. Magana, A.M., Quintanilla, R.: On the time decay of solutions in porous-elasticity with quasi-static microvoids. J. Math. Anal. Appl. 331(1), 617–630 (2007). https://doi.org/10.1016/j.jmaa.2006.08.086

    Article  MathSciNet  MATH  Google Scholar 

  17. Magana, A.M., Quintanilla, R.: On the time decay of solutions in one-dimensional theories of porous materials. Int. J. Solids Struct. 43, 3414–3427 (2006). https://doi.org/10.1016/j.ijsolstr.2005.06.077

    Article  MathSciNet  MATH  Google Scholar 

  18. Nicaise, S., Pignotti, C.: Stabilization of the wave equation with boundary or internal distributed delay. Diff Int Equ 21(9–10), 935–958 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Pazy, A.: Semigroups of linear operators and applications to partial differential equations, Applied Math. Sciences, Springer-Verlag, New York, 44 (1983)

  20. Pamplona, P.X., Munoz Rivera, J.E., Quintanilla, R.: Stabilization in elastic solids with voids, J. Math. Anal. Appl. 350 37–49. (2009). https://doi.org/10.1016/j.jmaa.2008.09.026

  21. Quintanilla, R.: Slow decay for one-dimensional porous dissipation elasticity. Appl. Math. Lett. 16(4), 487–491 (2003). https://doi.org/10.1016/S0893-9659(03)00025-9

    Article  MathSciNet  MATH  Google Scholar 

  22. Santos, M.L., Campelo, A.D.S., D. A. J. unior,: On the decay rates of porous elastic systems. J. Elast. 127(1), 79–101 (2017). https://doi.org/10.1007/s10659-016-9597-y

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Acknowledgements

The authors wish to thank deeply the anonymous referee for his/her useful remarks and his/her careful reading of the proofs presented in this paper.

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Authors and Affiliations

  1. university of 20 August 1955, Skikda, Algeria

    Fahima Hebhoub, Lamine Bouzettouta & Karima Ghennam

  2. École Normale Supérieure d’Enseignement Technologique, Skikda, Algeria

    Khoudir Kibech

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  1. Fahima Hebhoub
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  2. Lamine Bouzettouta
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  3. Karima Ghennam
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  4. Khoudir Kibech
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Correspondence to Lamine Bouzettouta.

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Hebhoub, F., Bouzettouta, L., Ghennam, K. et al. Stabilization of a Microtemperature Porous-Elastic System with Distributed Delay-Time. Mediterr. J. Math. 19, 222 (2022). https://doi.org/10.1007/s00009-022-02123-1

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  • DOI: https://doi.org/10.1007/s00009-022-02123-1

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