Global existence and exponential stability of a fractionally damped plate equation with past history

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This article was retracted on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 paper is an extension of our earlier research [1]. In this article, we discussed the fractionally damped plate equation with infinite memory. We proved that solutions are global under different initial and boundary conditions. Subsequently, we investigated the exponential stability of the solutions through Lyapunov functionals in order to establish stability conditions sufficient for stability analysis.

Keywords

plate equation / general decay / fractional derivatives / infinite memory / global existence / nonlinear equations

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This article has been retracted.Please see the Retraction Notice for more detail: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] M. F. Aslam, J. Hao, S. Boulaaras, L. Bashir, Blow-up of solutions in a fractionally damped plate equation with infinite memory and logarithmic nonlinearity, Axioms, 14 (2025), 80. https://doi.org/10.3390/axioms14020080

[2] E. Blanc, G. Chiavassa, B. Lombard, Boit-JKD model: Simulation of 1D transient poroelastic waves with fractional derivative, J. Comput. Phys., 237 (2013), 1-20. https://doi.org/10.1016/j.jcp.2012.12.003

[3] J. U. Choi, R. C. Maccamy, Fractional order Volterra equations with applications to elasticity, J. Math. Anal. Appl., 139 (1989), 448-464. https://doi.org/10.1016/0022-247x(89)90120-0

[4] R. L. Magin, Fractional Calculus in Bioengineering , Begell House: Redding, CT, USA, 2006.

[5] V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media , Springer Science & Business Media: Berlin/Heidelberg, Germany, 2011.

[6] D. Valerio, J. A. T. Machado, V. Kiryakova, Some pioneers of the applications of fractional calculus, Fract. Calc. Appl. Anal., 17 (2014), 552-578.

[7] J. F. G. Aguilar, D. Baleanu, Solutions of the telegraph equations using a fractional calculus approach, Proc. Rom. Acad. Ser. A, 15 (2014), 27-34.

[8] J. A. T. Machado, A. M. Lopes, Analysis of natural and artificial phenomena using signal processing and fractional calculus, Fract. Calc. Appl. Anal., 18 (2015), 459-478.

[9] A. Benaissa, H. Benkhedda, Global existence and energy decay of solutions to a wave equation with a dynamic boundary dissipation of fractional derivative type, Z. Anal. Anwend., 37 (2018), 315-339. https://doi.org/10.4171/ZAA/1616

[10] R. Aounallah, S. Boulaaras, A. Zarai, B. Cherif, General decay and blow-up of solutions for a nonlinear wave equation with a fractional boundary damping, Math. Methods Appl. Sci., 43 (2020), 7175-7193. https://doi.org/10.1002/mma.6455

[11] S. Boulaaras, F. Kamache, Y. Bouizem, R. Guefaifia, General decay and blow-up of solutions for a nonlinear wave equation with memory and fractional boundary damping terms, Bound. Value Probl., 2020 (2020). https://doi.org/10.1186/s13661-020-01470-w

[12] B. Mbodje, Wave energy decay under fractional derivative controls, IMA J. Math. Control Inf., 23 (2006), 237-257. https://doi.org/10.1093/imamci/dni056

[13] N. Doudi, S. Boulaaras, N. Mezouar, R. Jan, Global existence, general decay and blow-up for a nonlinear wave equation with logarithmic source term and fractional boundary dissipation, Discret. Contin. Dyn. Syst.-S, 16 (2023), 1323-1345. http://dx.doi.org/10.3934/dcdss.2022106

[14] R. Aounallah, A. Benaissa, A. Zarai, Blow-up and asymptotic behavior for a wave equation with a time delay condition of fractional type, in Rendiconti del Circolo Matematico di Palermo Series 2, Springer: Berlin/Heidelberg, Germany, 2020. https://doi.org/10.1007/s12215-020-00545-y

[15] M. F. Aslam, J. Hao, Nonlinear logarithmic wave equations: Blow-up phenomena and the influence of fractional damping, infinite memory, and strong dissipation, Evol. Equ. Control Theory, 13 (2024), 1423-1435. https://doi.org/10.3934/eect.2024034

[16] M. F. Aslam, J. Hao, Z. Hajjej, L. Bashir, On the global existence, exponential decay and blow-up of a nonlinear wave equation subject to a boundary fractional damping and time-varying delay, Discrete Contin. Dyn. Syst. - Ser. S, 2025 (2025). https://doi.org/10.3934/dcdss.2025038

[17] Z. Hajjej, A suspension bridges with a fractional time delay: Asymptotic behavior and Blow-up in finite time, AIMS Math., 9 (2024), 22022-22040. http://dx.doi.org/2010.3934/math.20241070

[18] J. Lagnese, Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping, Int. Ser. Numer. Math., 1989 (1989), 211-236.

[19] J. M. Rivera, E. C. Lapa, R. Barreto, Decay rates for viscoelastic plates with memory, J. Elast., 44 (1996), 61-87. https://doi.org/10.1007/BF00042192

[20] V. Komornik, On the nonlinear boundary stabilization of Kirchhoff plates, NoDEA, 1 (1994), 323-337. https://doi.org/10.1007/BF01194984

[21] S. A. Messaoudi, Global existence and nonexistence in a system of Petrovsky, J. Math. Anal. Appl., 265 (2002), 296-308. https://doi.org/10.1006/jmaa.2001.7697

[22] W. Chen, Y. Zhou, Global nonexistence for a semilinear Petrovsky equation, Nonlinear Anal. A, 70 (2009), 3203-3208. https://doi.org/10.1016/j.na.2008.04.024

[23] S. A. Messaoudi, S. E. Mukiawa, Existence and decay of solutions to a viscoelastic plate equation, Electron. J. Differ. Equations, 2016 (2016), 1-14.

[24] M. Cavalcanti, M. Domingos, V. N. Cavalcanti, J. Ferreira, Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Methods Appl. Sci., 24 (2001), 1043-1053. https://doi.org/10.1002/mma.250

[25] 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

[26] M. I. Mustafa, M. Kafini, Decay rates for memory-type plate system with delay and source term, Math. Methods Appl. Sci., 40 (2017), 883-895. https://doi.org/10.1002/mma.4015

[27] M. M. Al-Gharabli, A. Guesmia, S. A. Messaoudi, Some existence and exponential stability results for a plate equation with strong damping and a logarithmic source term, in Differential Equations and Dynamical Systems, Springer: Berlin/Heidelberg, Germany, (2022), 1-15. https://doi.org/10.1007/s12591-022-00625-8

[28] M. M. Al-Gharabli, Stability results for a system of nonlinear viscoelastic plate equations with nonlinear frictional damping and logarithmic source terms, J. Dyn. Control. Syst., 30 (2024), 3. https://doi.org/10.1007/s10883-023-09676-8

[29] M. M. Al-Gharabli, A. M. Almahdi, M. Noor, J. D. Audu, Numerical and theoretical stability study of a viscoelastic plate equation with nonlinear frictional damping term and a logarithmic source term, Math. Comput. Appl., 27 (2022), 10. https://doi.org/10.3390/mca27010010

[30] M. M. Al-Gharabli, A. M. Al-Mahdi, Existence and stability results of a plate equation with nonlinear damping and source term, Electron. Res. Arch., 30 (2022), 4038-4065. http://dx.doi.org/10.3934/era.2022205

[31] M. D’Abbicco, L. G. Longen, The interplay between fractional damping and nonlinear memory for the plate equation, Math. Methods Appl. Sci., 45 (2022), 6951-6981. https://doi.org/10.1002/mma.8219

[32] J. E. M. Rivera, H. D. F. Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502. https://doi.org/10.1016/j.jmaa.2007.07.012

[33] P. X. Pamplona, J. E. M. Rivera, R. Quintanilla, On the decay of solutions for porous-elastic systems with history, J. Math. Anal. Appl., 379 (2011), 682-705. https://doi.org/10.1016/j.jmaa.2011.01.045

[34] Q. Peng, Z. Zhang, Stabilization and blow-up in an infinite memory wave equation with logarithmic nonlinearity and acoustic boundary conditions, J. Syst. Sci. Complexity, 37 (2024), 1368-1391. https://doi.org/10.1007/s11424-024-3132-1