General decay for a coupled wave problem with Lord-Shulman thermal heat law and delay

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

This article study a coupled one-dimensional hyperbolic system with Lord-Shulman thermal heat law on the first equation, and a general weak inherent damping, along with time-dependent coefficient and time-varying delay term on the second equation. Using the multiplier method, through a suitable Lyapunov functional, we prove a general decay stability result that encompasses both exponential and polynomial decay estimates as special cases. This result extends and generalizes existing findings in the literature related to the system under study.

Keywords

general decay / coupled wave / Lord-Shulman thermal law / time varying coefficient and delay

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] H. Bounadja, O. Benhamouda, S. Messaoudi, A stability result of a one-dimensional coupled system of wave equations with Fourier heat conduction, Ann. Univ. Ferrara, 71 (2025), 34. https://doi.org/10.1007/s11565-025-00591-3

[2] T. A. Apalara, H. A. Almutairi, Well-posedness and stability of doublewall carbon nanotubes Timoshenko system under Lord-Shulman thermoelasticity, Discrete Contin. Dyn. Syst. - Ser. S, 2025. https://doi.org/10.3934/dcdss.2025020

[3] C. Raposo, L. Correia, J. Ribeiro, A. Cunha, Suspension bridge with internal damping, Acta Mech., 235 (2024), 203-214. https://doi.org/10.1007/s00707-023-03744-7

[4] S. E. Mukiawa, Decay result for a delay viscoelastic plate equation, Bull. Braz. Math. Soc., 51 (2020), 333-356. https://doi.org/10.1007/s00574-019-00155-y

[5] J. P. Richard, Time-delay systems: an overview of some recent advances and open problems, Automatica, 39 (2003), 1667-1694. https://doi.org/10.1016/S0005-1098(03)00167-5

[6] M. J. Lee, J. R. Kang, General stability for the viscoelastic wave equation with nonlinear time- varying delay, Mathematics, 22 (2023), 4593. https://doi.org/10.3390/math11224593

[7] A. Braik, S. M. Mirgani, E. I. Hassan, K. Zennir, Well-posedness and energy decay rates for a timoshenko-type system with internal time-varying delay in the displacement, Symmetry, 10 (2023), 1878. https://doi.org/10.3390/sym15101878

[8] J. D. Audu, S. E. Mukiawa, D. S. A. Júnior, General decay estimate for a two-dimensional plate equation with time-varying feedback and time-varying coefficient, Results Appl. Math., 12 (2021), 100219. https://doi.org/10.1016/j.rinam.2021.100219

[9] Z. Liu, R. Peng, Q. Zhang, Polynomial stability of the Rao-Nakra beam with a single internal viscous damping, J. Differ. Equations, 269 (2020), 6125-6162. https://doi.org/10.1016/j.jde.2020.04.030

[10] S. E. Mukiawa, Stability result of a suspension bridge Problem with time-varying delay and time-varying weight, Arabian J. Math., 10 (2021), 659-668. https://doi.org/10.1007/s40065-021-00345-x

[11] S. E. Mukiawa, The effect of time-varying delay damping on the stability of porous elastic system, Open J. Math. Sci., 5 (2021), 147-161. https://doi.org/10.30538/oms2021.0152

[12] F. Alabau, Stabilisation frontiere indirecte de systemes faiblement couplés Indirect boundary stabilization of weakly coupled systems, C.R. Acad. Sci. Série. I: Math., 328 (1999), 1015-1020. https://doi.org/10.1016/S0764-4442(99)80316-4

[13] M. L. Santos, M. P. C. Rocha, S. C. Gomes, Polynomial stability of a coupled system of wave equations weakly dissipative, Appl. Anal., 86 (2007), 1293-1302. https://doi.org/10.1080/00036810701624785

[14] F. Lobato, S. Cordeiro, M. L. Santos, A. Júnior, Optimal polynomial decay to coupled wave equations and its numerical properties, J. Appl. Math., (2014), 897080. https://doi.org/10.1155/2014/897080

[15] A. Guesmia, S. A. Messaoudi, M. Zahri, General decay of solutions of a weakly coupled abstract evolution equation with one finite memory control, J. Appl. Anal. Comput., 14 (2024), 3539-3557. https://doi.org/10.11948/20240081

[16] R. G. C. Almeida, M. L. Santos, Lack of exponential decay of a coupled system of wave equations with memory, Nonlinear Anal. Real World Appl., 12 (2011), 1023-1032. https://doi.org/10.1016/j.nonrwa.2010.08.025

[17] S. E. Mukiawa, C. D. Enyi, S. A. Messaoudi, Stability of thermoelastic Timoshenko beam with suspenders and time-varying feedback, Adv. Contin. Discrete Models, 2023 (2023), 7. https://doi.org/10.1186/s13662-023-03752-w

[18] S. Nicaise, C. Pignotti Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585. https://doi.org/10.1137/060648891

[19] A. Benaissa, A. Benaissa, S. A. Messaoudi, Global existence and energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks, J. Math. Phys., 53 (2012), 123514. https://doi.org/10.1063/1.4765046

[20] V. I. Arnold, Mathematical Methods of Classical Mechanics , Springer-Verlag, New York, 1989.

[21] I. Lasiecka, D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differ. Integr. Equations, 6 (1993), 507-533. https://doi.org/10.57262/die/1370378427