Global attractors for a class of viscoelastic plate equations with past history

Quan Zhou; Yang Liu; Dong Yang; Quan Zhou; Yang Liu; Dong Yang

doi:10.3934/math.20241212

AIMS Mathematics

2024, Volume 9, Issue 9: 24887-24907. doi: 10.3934/math.20241212

Previous Article Next Article

Research article

Global attractors for a class of viscoelastic plate equations with past history

1.
College of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, China
2.
College of Creative Design, Chongqing City Vocational College, Chongqing 402160, China

Received: 27 June 2024 Revised: 06 August 2024 Accepted: 12 August 2024 Published: 26 August 2024
MSC : 35B41, 35L75, 37N15

This paper is concerned with a class of viscoelastic plate equations with past history. We first transform the original initial-boundary value problem into an equivalent one by means of the history space framework. Then we use the perturbed energy method to establish a stabilizability estimate. By employing the gradient property and quasi-stability of the dynamical system, we obtain the existence of a global attractor with finite fractal dimension.
- viscoelastic plate equations,
- past history,
- global attractors,
- gradient dynamical system,
- quasi-stability
Citation: Quan Zhou, Yang Liu, Dong Yang. Global attractors for a class of viscoelastic plate equations with past history[J]. AIMS Mathematics, 2024, 9(9): 24887-24907. doi: 10.3934/math.20241212

Related Papers:

Abstract

This paper is concerned with a class of viscoelastic plate equations with past history. We first transform the original initial-boundary value problem into an equivalent one by means of the history space framework. Then we use the perturbed energy method to establish a stabilizability estimate. By employing the gradient property and quasi-stability of the dynamical system, we obtain the existence of a global attractor with finite fractal dimension.

References

[1]	J. E. Lagnese, Boundary stabilization of thin plates, Philadelphia, PA: SIAM, 1989. https://doi.org/10.1137/1.9781611970821
[2]	J. E. Lagnese, J. L. Lions, Modelling analysis and control of thin plates, Paris: Masson, 1988.
[3]	C. Giorgi, F. M. Vegni, Uniform energy estimates for a semilinear evolution equation of the Mindlin-Timoshenko beam with memory, Math. Comput. Model., 39 (2004), 1005–1021. https://doi.org/10.1016/S0895-7177(04)90531-6 doi: 10.1016/S0895-7177(04)90531-6
[4]	W. Weaver, S. P. Timoshenko, D. H. Young, Vibration problems in engineering, 5 Eds., New York: John Wiley and Sons, 1990.
[5]	M. Fabrizio, C. Giorgi, V. Pata, A new approach to equations with memory, Arch. Rational Mech. Anal., 198 (2010), 189–232. https://doi.org/10.1007/S00205-010-0300-3 doi: 10.1007/S00205-010-0300-3
[6]	F. Alabau-Boussouira, P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Math., 347 (2009), 867–872. https://doi.org/10.1016/J.CRMA.2009.05.011 doi: 10.1016/J.CRMA.2009.05.011
[7]	J. E. Muñoz Rivera, E. C. Lapa, R. Barreto, Decay rates for viscoelastic plates with memory, J. Elasticity, 44 (1996), 61–87. https://doi.org/10.1007/BF00042192 doi: 10.1007/BF00042192
[8]	J. E. Muñoz Rivera, L. H. Fatori, Smoothing effect and propagations of singularities for viscoelastic plates, J. Math. Anal. Appl., 206 (1997), 397–427. https://doi.org/10.1006/JMAA.1997.5223 doi: 10.1006/JMAA.1997.5223
[9]	M. M. Cavalcanti, V. N. Domingos Cavalcanti, T. F. Ma, Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains, Differ. Integral Equ., 17 (2004), 495–510. https://doi.org/10.57262/DIE/1356060344 doi: 10.57262/DIE/1356060344
[10]	F. Alabau-Boussouira, P. Cannarsa, D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254 (2008), 1342–1372. https://doi.org/10.1016/J.JFA.2007.09.012 doi: 10.1016/J.JFA.2007.09.012
[11]	P. Cannarsa, D. Sforza, Integro-differential equations of hyperbolic type with positive definite kernels, J. Differ. Equations, 250 (2011), 4289–4335. https://doi.org/10.1016/J.JDE.2011.03.005 doi: 10.1016/J.JDE.2011.03.005
[12]	Z. Hajjej, General decay of solutions for a viscoelastic suspension bridge with nonlinear damping and a source term, Z. Angew. Math. Phys., 72 (2021), 90. https://doi.org/10.1007/S00033-021-01526-6 doi: 10.1007/S00033-021-01526-6
[13]	M. A. Jorge Silva, J. E. Muñoz Rivera, R. Racke, On a class of nonlinear viscoelastic Kirchhoff plates: well-posedness and general decay rates, Appl. Math. Optim., 73 (2016), 165–194. https://doi.org/10.1007/S00245-015-9298-0 doi: 10.1007/S00245-015-9298-0
[14]	E. H. Gomes Tavares, M. A. Jorge Silva, T. F. Ma, Sharp decay rates for a class of nonlinear viscoelastic plate models, Commun. Contemp. Math., 20 (2018), 1750010. https://doi.org/10.1142/S0219199717500109 doi: 10.1142/S0219199717500109
[15]	M. A. Jorge Silva, T. F. Ma, On a viscoelastic plate equation with history setting and perturbation of $p$-Laplacian type, IMA J. Appl. Math., 78 (2013), 1130–1146. https://doi.org/10.1093/IMAMAT/HXS011 doi: 10.1093/IMAMAT/HXS011
[16]	D. C. Pereira, G. M. Araújo, C. A. Raposo, V. R. Cabanillas, Blow-up results for a viscoelastic beam equation of $p$-Laplacian type with strong damping and logarithmic source, Math. Method. Appl. Sci., 46 (2023), 8831–8854. https://doi.org/10.1002/mma.9020 doi: 10.1002/mma.9020
[17]	A. Merah, F. Mesloub, On a viscoelastic plate equation with a polynomial source term and $\overrightarrow{p}(x, t)$-Laplacian operator in the presence of delay term, J. Innov. Appl. Math. Comput. Sci., 2 (2022), 92–107.
[18]	R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, 3 Eds., New York: Springer, 1997. https://doi.org/10.1007/978-1-4612-0645-3
[19]	A. V. Babin, M. I. Vishik, Attractors of evolution equations, Amsterdam: North-Holland Publishing Co., 1992.
[20]	O. Ladyzhenskaya, Attractors for semigroups and evolution equations, Cambridge: Cambridge University Press, 1991. https://doi.org/10.1017/9781009229814
[21]	J. K. Hale, Asymptotic behavior of dissipative systems, Providence, RI: American Mathematical Society, 1988. https://doi.org/10.1090/SURV/025
[22]	Q. F. Ma, S. H. Wang, C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541–1570. https://doi.org/10.1512/IUMJ.2002.51.2255 doi: 10.1512/IUMJ.2002.51.2255
[23]	X. M. Peng, Y. D. Shang, H. F. Di, Long-time dynamics for a nonlinear viscoelastic Kirchhoff plate equation, Chin. Ann. Math. Ser. B, 41 (2020), 627–644. https://doi.org/10.1007/S11401-020-0222-9 doi: 10.1007/S11401-020-0222-9
[24]	I. Chueshov, I. Lasiecka, Von Karman evolution equations, New York: Springer, 2010. https://doi.org/10.1007/978-0-387-87712-9
[25]	B. Feng, M. A. Jorge Silva, A. H. Caixeta, Long-time behavior for a class of semi-linear viscoelastic Kirchhoff beams/plates, Appl. Math. Optim., 82 (2020), 657–686. https://doi.org/10.1007/S00245-018-9544-3 doi: 10.1007/S00245-018-9544-3
[26]	G. W. Liu, M. A. Jorge Silva, Attractors and their properties for a class of Kirchhoff models with integro-differential damping, Appl. Anal., 101 (2022), 3284–3307. https://doi.org/10.1080/00036811.2020.1846722 doi: 10.1080/00036811.2020.1846722
[27]	M. Conti, V. Danese, C. Giorgi, V. Pata, A model of viscoelasticity with time-dependent memory kernels, Amer. J. Math., 140 (2018), 349–389. https://doi.org/10.1353/AJM.2018.0008 doi: 10.1353/AJM.2018.0008
[28]	F. Di Plinio, V. Pata, S. Zelik, On the strongly damped wave equation with memory, Indiana Univ. Math. J., 57 (2008), 757–780. https://doi.org/10.1512/IUMJ.2008.57.3266 doi: 10.1512/IUMJ.2008.57.3266
[29]	Y. Liu, Global attractors for a nonlinear plate equation modeling the oscillations of suspension bridges, Commun. Anal. Mech., 15 (2023), 436–456. https://doi.org/10.3934/CAM.2023021 doi: 10.3934/CAM.2023021
[30]	Y. Liu, L. Zhang, On a viscoelastic Kirchhoff equation with fractional Laplacian, Discrete Contin. Dyn. Syst. Ser. S, 17 (2024), 2543–2565. https://doi.org/10.3934/DCDSS.2024007 doi: 10.3934/DCDSS.2024007

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)