SpringerLink
Log in

Stabilization of swelling porous elastic soils with fluid saturation, time varying-delay and time-varying weights

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this article we study the well-posedness and exponential stability to the one-dimensional system in the linear isothermal theory of swelling porous elastic soils subject with time-varying weights and time-varying delay. We prove existence of global solution for the problems combining semigroup theory with the Kato’s variable norm technique. To prove exponential stability, we apply the energy method without the equal wave speeds assumption.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Apalara, T.A.: Asymptotic behavior of weakly dissipative Timoshenko system with internal constant delay feedbacks. Appl. Anal. 95(1), 187–202 (2016)

    Article  MathSciNet  Google Scholar 

  2. Barros, V., Nonato, C., Raposo, C.: Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights. Electron. Res. Arch. 28, 549–567 (2020)

    Article  MathSciNet  Google Scholar 

  3. Benaissa, A., Benguessoum, A., Messaoudi, S.A.: Energy decay of solutions for a wave equation with a constant weak delay and a weak internal feedback. Electron. J. Qual. Theory Differ. Equ. 11, 1–13 (2014)

    Article  MathSciNet  Google Scholar 

  4. Datko, R.: Not All Feedback Stabilized Hyperbolic Systems are Robust with Respect to Small Time Delays in Their Feedbacks. SIAM J. Control Optim. 26(3), 697–713 (1988)

    Article  MathSciNet  Google Scholar 

  5. Datko, R., Lagnese, J., Polis, M.P.: An example of the effect of time delays in boundary feedback stabilization of wave equations. In: 1985 24th IEEE Conference on Decision and Control. Fort Lauderdale, FL, USA, pp. 117 (1985)

  6. Dos Santos, M.J., Feng, B., Almeida Júnior, D.S., Santos, M.L.: Global and exponential attractors for a nonlinear porous elastic system with delay term. Discret. Contin. Dyn. Syst. B 22(11), 1–24 (2020)

    MATH  Google Scholar 

  7. Dos Santos, M.J., Freitas, M.M, Ramos, A.J.A., Almeida Júnior, D.S., Rodrigues, L.R.S.: Long-time dynamics of a nonlinear Timoshenko beam with discrete delay term and nonlinear dam**. J. Math. Phys. 61(061505), 1–17 (2020)

  8. Feng, B., Pelicer, M.L.: Global existence and exponential stability for a nonlinear Timoshenko system with delay. Bound. Value Probl. 2015(1), 206 (2017)

    Article  Google Scholar 

  9. Feng, B., Yang, X.: Long-time dynamics for a nonlinear Timoshenko system with delay. Appl. Anal. 96(4), 606–625 (2017)

    Article  MathSciNet  Google Scholar 

  10. Kato, T.: Abstract differential equations and nonlinear mixed problems. Publications of the Scuola Normale Superiore, Edizioni della Normale (1998)

    Google Scholar 

  11. Kato, T.: Linear and Quasi-Linear Equations of Evolution of Hyperbolic Type. Springer (2011)

  12. Kirane, M., Said-Houari, B.: Existence and asymptotic stability of a viscoelastic wave equation with a delay. Z. Angew. Math. Phys. 6, 1065–1082 (2011)

    Article  MathSciNet  Google Scholar 

  13. Kirane, M., Said-Houari, B., Anwar, M.N.: Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Commun. Pure Appl. Anal. 10(2), 667–686 (2011)

    Article  MathSciNet  Google Scholar 

  14. 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. Continuum Mech. Therm. 3, 731–746 (2017)

    Article  MathSciNet  Google Scholar 

  15. Nicaise, S., Pignotti, C.: Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45(5), 1561–1585 (2006)

    Article  MathSciNet  Google Scholar 

  16. Nicaise, S., Pignotti, C.: Interior feedback stabilization of wave equations with time dependence delay. Electron. J. Diff. Equ. 41, 1–20 (2011)

    MATH  Google Scholar 

  17. Nicaise, S., Pignotti, C., Valein, J.: Exponential stability of the wave equation with boundary time-varying delay. Discrete Contin. Dyn. Syst. S 4(3), 693–722 (2011)

    Article  MathSciNet  Google Scholar 

  18. Park, S.: Long-time behavior for suspension bridge equations with time delay. Z. Angew. Math. Phys. 69(45), 1–12 (2018)

    MathSciNet  MATH  Google Scholar 

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

  20. Ramos, A.J.A., Dos Santos, M.J., Freitas, M.M., Almeida Júnior, D.S.: Existence of attractors for a nonlinear Timoshenko system with delay. J. Dyn. Differ. Equ. 1–24 (2019)

  21. Raposo, C.A., Apalara, T.A., Ribeiro, J.O.: Analyticity to transmission problem with delay in porous-elasticity. J. Math. Anal. Appl. 466(1), 819–834 (2018)

    Article  MathSciNet  Google Scholar 

  22. Raposo, C.A., Chuquipoma, J.A.D., Avila, J.A.J., Santos, M.L.: Exponential decay and numerical solution for a Timoshenko system with delay term in the internal feedbacky. Int. J. Anal. Appl. 3(1), 1–13 (2013)

    MATH  Google Scholar 

  23. Said-Houari, B., Laskri, Y.: A stability result of a Timoshenko system with a delay term in the internal feedback. Appl. Math. Comput. 217(6), 2857–2869 (2010)

    MathSciNet  MATH  Google Scholar 

  24. Xu, G.Q., Yung, S.P., Li, L.K.: Stabilization of wave systems with input delay in the boundary control. ESAIM Control Optim. Calc. Var. 12(4), 770–785 (2006)

    Article  MathSciNet  Google Scholar 

  25. Yang, X., Zhang, J., Lu, Y.: Dynamics of the nonlinear Timoshenko system with variable delay. Appl. Math. Optim. 83, 297–326 (2021)

    Article  MathSciNet  Google Scholar 

  26. Atkin, R.J., Craine, R.E.: Continuum theories of mixtures: basic theory and historical development. Q. J. Mech. Appl. Math. 29(2), 209–244 (1976)

    Article  MathSciNet  Google Scholar 

  27. Klika, V.: A guide through available mixture theories for applications. Crit. Rev. Solid State Mater. Sci. 39(2), 154–174 (2013)

    Article  Google Scholar 

  28. Bedford, A., Drumheller, D.: Theories of immiscible and structured mixtures. Int. J. Eng. Sci. 21(8), 863–960 (1983)

    Article  MathSciNet  Google Scholar 

  29. Eringen, A.C.: A continuum theory of swelling porous elastic soils. Int. J. Eng. Sci. 32(8), 1337–1349 (1994)

    Article  MathSciNet  Google Scholar 

  30. Quintanilla, R.: Exponential stability for one-dimensional problem of swelling porous elastic soils with fluid saturation. J. Comput. Appl. Math. 145, 525–533 (2002)

    Article  MathSciNet  Google Scholar 

  31. Wang, J.-M., Guo, B.-Z.: On the stability of swelling porous elastic soils with fluid saturation by one internal dam**. IMA J. Appl. Math. 71, 565–582 (2006)

    Article  MathSciNet  Google Scholar 

  32. Quintanilla, R.: On the linear problem of swelling porous elastic soils. J. Math. Anal. Appl. 269(1), 50–72 (2002)

    Article  MathSciNet  Google Scholar 

  33. Quintanilla, R.: Exponential stability of solutions of swelling porous elastic soils. Meccanica. 39(1), 139–145 (2004)

    Article  MathSciNet  Google Scholar 

  34. Bofill, F.: Quintanilla, R: Anti-plane shear deformations of swelling porous elastic soils. Int. J. Eng. Sci. 41(8), 801–816 (2003)

    Article  MathSciNet  Google Scholar 

  35. Karabomi, S., Smith, Urai B., J., Heidug, W. and Oort, E.: The swelling of clays: Molecularsimulations of the hydration of montmorillonite. Science 271, 1102–1104 (1996)

  36. Wilcox, R.D.: Surface area approach key to borehole stability. Oil Gas J. 26, 66–80 (1990)

    Google Scholar 

  37. Wray, W.: So Your Home is Built on Expansive Soils. Amer. Soc. Civil Eng. (1995)

  38. Sridharan, A., Rao, G.V.: Mechanisms controlling the secondary compression of clays. Geotechnique. 32(3), 249–260 (1982)

    Article  Google Scholar 

  39. Hueckel, T.: Water mineral interaction in hygromechanics of clays exposed to environ-mental loads: a mixture theory approach. Geotech. J. 29 1071–1086

  40. Nonato, C., Dos Santos M. J. and Raposo, C.: Dynamics of Timoshenko system with time-varying weight and time-varying delay. Discret. Contin. Dyn. Syst. - B. (2021)

  41. Nonato, C., Raposo, C. and Feng, B.: Exponential stability for a thermoelastic laminated beam with nonlinear weights and time-varying delay. Asympt. Anal. 1–29. (2021)

  42. Wang, Sp., Ma, Qz. : Uniform attractors for the non-autonomous suspension bridge equation with time delay. J. Inequal. Appl. 180. (2019)

  43. Ramos, A.J.A., Freitas, M.M., Almeida, D.S., Jr., Noé, A.S., Dos Santos, M.J.: Stability results for elastic porous media swelling with nonlinear dam**. J. Math. Phys. 61, 101505 (2020)

    Article  MathSciNet  Google Scholar 

  44. Ramos, A.J.A., Almeida, D.S., Jr., Freitas, M.M., Noé, A.S., Dos Santos, M.J.: Stabilization of swelling porous elastic soils with fluid saturation and delay time terms. J. Math. Phys. 62, 021507 (2021)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the referees for their constructive remarks, which have enhanced the presentation of this paper.

Funding

C. A. S. Nonato thanks CAPES for funding the doctoral scholarship.

A. J. A. Ramos thanks the CNPq for financial support through the project grant number 310729/2019-0.

Author information

Authors and Affiliations

  1. Department of Mathematics, Federal University of Bahia, Salvador, Bahia, Brazil

    C. A. S. Nonato

  2. Faculty of Mathematics, Federal University of Pará, Salinópolis, Pará, Brazil

    A. J. A. Ramos & M. M. Freitas

  3. Department of Mathematics, Federal University of São João del-Rei, São João del-Rei, Minas Gerais, Brazil

    C. A. Raposo

  4. Faculty of Exact Sciences and Technology, Federal University of Pará, Abaetetuba, Pará, Brazil

    M. J. Dos Santos

Authors
  1. C. A. S. Nonato
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. J. A. Ramos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. A. Raposo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. J. Dos Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. M. M. Freitas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. J. Dos Santos.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nonato, C.A.S., Ramos, A.J.A., Raposo, C.A. et al. Stabilization of swelling porous elastic soils with fluid saturation, time varying-delay and time-varying weights. Z. Angew. Math. Phys. 73, 20 (2022). https://doi.org/10.1007/s00033-021-01648-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-021-01648-x

Keywords

Mathematics Subject Classification

Search

Navigation