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On the stabilization of linear porous elastic materials by microtemperature effect and porous damping

  • Hanni Dridi
  • Abdelhak DjebablaEmail author
Article
  • 9 Downloads

Abstract

In this paper, we study the asymptotic behavior of solutions for the porous thermoelastic system with temperatures and microtemperatures effects. Our main result is to prove the exponential stability in case of zero thermal conductivity and without any condition on the coefficients of the system. This result is new and improve some recent results in the literature.

Keywords

Exponential decay Porous system Microtemperature damping 

Mathematics Subject Classification

435B37 35L55 74D05 93D15 

Notes

References

  1. 1.
    Goodman, M.A., Cowin, S.C.: A continuum theory for granular materials. Arch. Ration. Mech. Anal. 44(4), 249–266 (1972)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ieşan, D., Quintanilla, R.: On a theory of thermoelasticity with microtemperatures. J. Therm. Stresses 23(3), 199–215 (2000)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ieşan, D.: On a theory of micromorphic elastic solids with microtemperatures. J. Therm. Stresses 24(8), 737–752 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ieşan, D.: Thermoelastic Models of Continua. Springer, Dordrecht (2004)CrossRefGoogle Scholar
  5. 5.
    Ieşan, D., Quintanilla, R.: A theory of porous thermoviscoelastic mixtures. J. Therm. Stresses 30(7), 693–714 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chiriţ a, S., Ciarletta, M., D’Apice, C.: On the theory of thermoelasticity with microtemperatures. J. Math. Anal. Appl. 397(1), 349–361 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Quintanilla, R.: Slow decay for one-dimensional porous dissipation elasticity. Appl. Math. Lett. 16(4), 487–491 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Casas, P.S., Quintanilla, R.: Exponential stability in thermoelasticity with microtemperatures. Int. J. Eng. Sci. 43(1–2), 33–47 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Liu, Z., Zheng, S.: Semigroups Associated with Dissipative Systems. Chapman and Hall, Boca Raton (1999)zbMATHGoogle Scholar
  10. 10.
    Casas, P.S., Quintanilla, R.: Exponential decay in one-dimensional porous-thermo-elasticity. Mech. Res. Commun. 32(6), 652–658 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Magaña, A., Quintanilla, R.: On the time decay of solutions in one-dimensional theories of porous materials. Int. J. Solids Struct. 43(11–12), 3414–3427 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Apalara, T.A.: Exponential decay in one-dimensional porous dissipation elasticity. Q. J. Mech. Appl. Math. 70(4), 553–572 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Apalara, T.A.: General decay of solutions in one-dimensional porous-elastic system with memory. J. Math. Anal. Appl. 469, 457–471 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fareh, A., Messaoudi, S.A.: Energy decay for a porous thermoelastic system with thermoelasticity of second sound and with a non-necessary positive definite energy. Appl. Math. Comput. 293, 493–507 (2017)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Muñoz Rivera, J.E., Quintanilla, R.: On the time polynomial decay in elastic solids with voids. J. Math. Anal. Appl. 338(2), 1296–1309 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Pamplona, P.X., Muñoz Rivera, J.E., Quintanilla, R.: Stabilization in elastic solids with voids. J. Math. Anal. Appl. 350(1), 37–49 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Pamplona, P.X., Muñoz Rivera, J.E., Quintanilla, R.: On the decay of solutions for porous-elastic systems with history. J. Math. Anal. Appl. 379(2), 682–705 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Santos, M.L., Junior, D.A.: On porous-elastic system with localized damping. Z. Angew. Math. Phys. 67(3), 1–18 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Santos, M.L., Campelo, A.D.S., Junior, D.A.: On the decay rates of porous elastic systems. J. Elast. 127(1), 79–101 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Santos, M.L., Junior, D.A.: On the porous-elastic system with Kelvin–Voigt damping. J. Math. Analy. Appl. 445(1), 498–512 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Soufyane, A.: Energy decay for porous-thermo-elasticity systems of memory type. Appl. Anal. 87(4), 451–464 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Soufyane, A., Afilal, M., Aouam, T., Chacha, M.: General decay of solutions of a linear one-dimensional porous-thermoelasticity system with a boundary control of memory type. Nonlinear Anal. 72(11), 3903–3910 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hachelfi, M., Djebabla, A., Tatar, N.: On the decay of the energy for linear thermoelastic systems by thermal and micro-temperature effects. Eurasian J. Math. Comput. Appl. 6(2018), 29–37 (2018)Google Scholar
  24. 24.
    Apalara, T.A.: On the stability of porous-elastic system with microtemparatures. J. Therm. Stresses 42(2), 265–278 (2019) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Liu, Z., Zheng, S.: Semigroups associated with dissipative systems. In: Pitman Research Notes in Mathematics, vol. 398. Chapman and Hall, Boca Raton (1999)Google Scholar
  26. 26.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)CrossRefGoogle Scholar

Copyright information

© Università degli Studi di Ferrara 2019

Authors and Affiliations

  1. 1.Laboratory of Applied MathematicsUniversity Badji MokhtarAnnabaAlgeria

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