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On the Stabilization of a Memory-Type Porous Thermoelastic System

  • Tijani A. ApalaraEmail author
Article
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Abstract

In this work, we consider a one-dimensional porous thermoelastic system with memory effects. We prove a general decay result, for which exponential and polynomial decay results are special cases, depending only on the kernel of the memory effects. Our result is established irrespective of the wave speeds of the system. The result obtained is new and improves previous results in the literature.

Keywords

Stability Asymptotic behavior of solutions Porous system Thermoelasticity Relaxation function 

Mathematics Subject Classification

35B35 35B40 93D20 

Notes

Acknowledgements

The author thanks UHB for its continuous support. The author would also like to thank the editors and anonymous referees for their helpful comments and suggestions.

References

  1. 1.
    Goodman, M.A., Cowin, S.C.: A continuum theory for granular materials. Arch. Ration. Mech. Anal. 44(4), 249–266 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Nunziato, J.W., Cowin, S.C.: A nonlinear theory of elastic materials with voids. Arch. Ration. Mech. Anal. 72(2), 175–201 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ieşan, D.: A theory of thermoelastic materials with voids. Acta Mech. 60(1–2), 67–89 (1986)Google Scholar
  4. 4.
    Ieşan, D.: On a theory of micromorphic elastic solids with microtemperatures. J. Thermal Stresses 24(8), 737–752 (2001)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ieşan, D.: Thermoelastic Models of Continua. Springer, Berlin (2004)zbMATHGoogle Scholar
  6. 6.
    Ieşan, D., Quintanilla, R.: A theory of porous thermoviscoelastic mixtures. J. Thermal Stresses 30(7), 693–714 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chiriţă, S., Ciarletta, M., D’Apice, C.: On the theory of thermoelasticity with microtemperatures. J. Math. Anal. Appl. 397(1), 349–361 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cowin, S.C., Nunziato, J.W.: Linear elastic materials with voids. J. Elast. 13(2), 125–147 (1983)CrossRefzbMATHGoogle Scholar
  9. 9.
    Cowin, S.C.: The viscoelastic behavior of linear elastic materials with voids. J. Elast. 15(2), 185–191 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Quintanilla, R., Ieşan, D.: On a theory of thermoelasticity with microtemperatures. J. Thermal stresses 23(3), 199–215 (2000)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Casas, P.S., Quintanilla, R.: Exponential decay in one-dimensional porous-thermo-elasticity. Mech. Res. Commun. 32(6), 652–658 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liu, Z., Zheng, S.: Semigroups Associated with Dissipative Systems. Boca, Chapman Hall/CRC (1999)zbMATHGoogle Scholar
  13. 13.
    Casas, P.S., Quintanilla, R.: Exponential stability in thermoelasticity with microtemperatures. Int. J. Eng. Sci. 43(1–2), 33–47 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pamplona, P.X., Muñoz Rivera, J.E., Quintanilla, R.: Stabilization in elastic solids with voids. J. Math. Anal. Appl. 350(1), 37–49 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    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. Nonl. Anal. 72(11), 3903–3910 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Quintanilla, R.: Slow decay for one-dimensional porous dissipation elasticity. Appl. Math. Lett. 16(4), 487–491 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Apalara, T.A.: Exponential decay in one-dimensional porous dissipation elasticity. Quart. J. Mech. Appl. Math. 70(4), 363–372 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Apalara, T.A.: General decay of solutions in one-dimensional porous-elastic system with memory. J. Math. Anal. Appl. (2017).  https://doi.org/10.1016/j.jmaa.2017.08.007
  19. 19.
    Santos, M.L., Jùnior, D.A.: On porous-elastic system with localized damping. Z. Angew. Math. Phys 67(3), 1–18 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Santos, M.L., Campelo, A.D.S., Jùnior, D.A.: Rates of decay for porous elastic system weakly dissipative. Acta Applicandae Mathematicae 151(1), 1–26 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Soufyane, A.: Energy decay for Porous-thermo-elasticity systems of memory type. Appl. Anal. 87(4), 451–464 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Almeida Júnior, D.S., Santos, M.L., Múnoz Rivera, J.E.: Stability to 1-D thermoelastic Timoshenko beam acting on shear force. Z. Angew. Math. Phys. 65(6), 1233–1249 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Messaoudi, S.A., Fareh, A.: General decay for a porous thermoelastic system with memory: the case of equal speeds. Nonlinear Analysis: TMA 74(18), 6895–6906 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Messaoudi, S.A., Fareh, A.: General decay for a porous thermoelastic system with memory: the case of nonequal speeds. Acta Math. Sci. 33(1), 23–40 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Apalara, T.A.: General stability of memory-type thermoelastic Timoshenko beam acting on shear force. Cont. Mech. Thermo. 30(2), 291–300 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of Hafr Al-Batin (UHB)Hafr Al-BatinSaudi Arabia

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