Discontinuity, Nonlinearity, and Complexity 13(4) (2024) 707--731 | DOI:10.5890/DNC.2024.12.010

Meriem Chabekh$^1$, Nadhir Chougui${}^1$, Fares Yazid$^2$, Abdelkader Saadallah$^1$

$^1$ Laboratory of Applied Mathematics, Ferhat Abbas S'etif 1 University, S'etif 19000, Algeria

$^2$ Laboratory of pure and applied Mathematics, Amar Teledji University, Laghouat 03000, Algeria

Abstract

In this paper, we consider a one-dimensional porous thermoelastic system with microtemperatures effects, past history term acting only on the porous equation and a delay term in the internal feedback. Under an appropriate assumptions on the kernel and between the weight of the delay and the weight of the damping, we prove the well-posedness of the system. Furthermore, we establish a general decay rate result for the energy, which allows a wider class of relaxation functions, and thus generalize some results in the literature. Finally, some numerical experiments are presented.

References

1. [1]	Goodman, M.A. and Cowin, S.C. (1972), A continuum theory for granular materials, Archive for Rational Mechanics and Analysis, 44(4), 249–266.

2. [2]	Cowin, S.C. and Nunziato, J.W. (1983), Linear elastic materials with voids, Journal of Elasticity, 13, 125–147.

3. [3]	Ie\c{s}an, D. (2004), Thermoelastic Models of Continua, Springer.

4. [4]	Ie\c{s}an, D. (1986), A theory of thermoelastic materials with voids, Acta Mechanica, 60, 67–89.

5. [5]	Ie\c{s}an, D. (2001), On a theory of micromorphic elastic solids with microtemperatures, Journal of Thermal Stresses, 24, 737–752.

6. [6]	Ie\c{s}an, D. and Quintanilla, R. (2000), On a theory of thermoelasticity with microtemperatures, Journal of Thermal Stresses, 23(3), 199-215.

7. [7]	Quintanilla, R. (2003), Slow decay for one-dimensional porous dissipation elasticity, Applied Mathematics Letters, 16(4), 487–491.

8. [8]	Casas, P.S. and Quintanilla, R. (2005), Exponential decay in one-dimensional porous-thermo-elasticity, Mechanics Research Communications(6), 32, 652–658.

9. [9]	Casas, P.S. and Quintanilla, R. (2005), Exponential stability in thermoelasticity with microtemperatures, International Journal of Engineering Science, 43, 33–47.

10. [10]	Maga{n}a, A. and Quintanilla, R. (2006), On the time decay of solutions in one-dimensional theories of porous materials, International Journal of Solids and Structures, 43, 3414–3427.

11. [11]	Datko, R., Lagnese, J., and Polis, M.P. (1986), An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM Journal on Control and Optimization, 24, 152–156.

12. [12]	Nicaise, S. and Pignotti, C. (2006), Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM Journal on Control and Optimization, 45(5), 1561–1585.

13. [13]	Kirane, M. and Said-Houari, B. (2011), Existence and asymptotic stability of a viscoelastic wave equation with a delay, Zeitschrift für angewandte Mathematik und Physik, 62(6), 1065–1082.

14. [14]	Said-Houari, B. and Laskri, Y. (2010), A stability result of a Timoshenko system with a delay term in the internal feedback, Applied Mathematics and Computation, 217, 2857–2869.

15. [15]	Guesmia, A. and Messaoudi, S.A. (2008), On the control of a viscoelastic damped Timoshenko-type system, Applied Mathematics and Computation, 206(2), 589–597.

16. [16]	Liu, W.J., Chen, K.W., and Yu, J. (2015), Existence and general decay for the full von Kármán beam with a thermo-viscoelastic damping, frictional dampings and a delay term, IMA Journal of Mathematical Control and Information, in press. URL: https://doi.org/10.1093/imamci/dnv056.

17. [17]	Messaoudi, S.A. and Apalara, T.A. (2014), General stability result in a memory type porous thermoelasticity system of type III, Arab Journal of Mathematical Sciences, 20(2), 213-232.

18. [18]	Mu\~{n}oz Rivera, J.E. and Fern{a}ndez Sare, H.D. (2008), Stability of Timoshenko systems with past history, Journal of Mathematical Analysis and Applications, 339(1), 482-502.

19. [19]	Pamplona, P.X., Mu\~{n}noz Rivera, J.E., and Quintanilla, R. (2011), On the decay of solutions for porous-elastic systems with history, Journal of mathematical analysis and applications, 379(2), 682-705.

20. [20]	Bernardi, C. and Copetti, M. I. M. (2016), Discretization of a nonlinear dynamic thermoviscoelastic Timoshenko beam model, ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 97(5), 1–18.

21. [21]	Chebbi, S. and Makram, H. (2019), Discrete energy behavior of a damped Timoshenko system, Computational and Applied Mathematics, 39(2020), 1-19.

22. [22]	EL Arwadi, T., Copetti, M.I.M., and Youssef, W. (2019), On the theoretical and numerical stability of the thermoviscoelastic Bresse system, ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 99(10), p.e201800207.

23. [23]	Khochemane, H.E. (2021), General stability result for a porous thermoelastic system with infinite history and microtemperatures effects, Mathematical Methods in the Applied Sciences, 45(3), 1538-1557.