Stability analysis of partially viscoelastic Reissner-Mindlin-Timoshenko plates

Cícero L. Frota; Marcio A. Jorge Silva; Sandro B. Pinheiro

doi:10.3934/eect.2025005

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2025, Volume 14, Issue 4: 748-781. Doi: 10.3934/eect.2025005

This issue Previous Article Global behavior of solutions to the nonlocal in time reaction-diffusion equations Next Article Evolution families in the framework of maximal regularity

Stability analysis of partially viscoelastic Reissner-Mindlin-Timoshenko plates

1.
Department of Mathematics, State University of Maringá, Maringá 87020-900, Paraná, Brazil
2.
Department of Mathematics, State University of Londrina, Londrina 86057-970, Paraná, Brazil

^*Corresponding author: Marcio A. Jorge Silva
^*Corresponding author: Marcio A. Jorge Silva

Received: July 2024

Revised: November 2024

Early access: January 2025

Published: August 2025

The second author is supported by the CNPq Grant 309929/2022-9 and Fundação Araucária Grant 226/2022. The third authors is supported by the CAPES, Finance Code 001.

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Abstract

This paper investigates the asymptotic stability of a partially viscoelastic Reissner-Mindlin-Timoshenko (RMT) plate system, incorporating the Boltzmann principle in the in-plane strain tensor related to rotation angles. Our main mathematical results include: $ (i) $ the asymptotic stability of solutions independent of any relations on the coefficients, using refined contradiction arguments through spectral analysis; $ (ii) $ additionally, we establish qualitative uniform and semi-uniform decay rates under the equal wave speeds assumption, achieved through direct estimates with new multipliers and the resolvent equation. These findings highlight the intricate nature of the bi-dimensional RMT-system and the role of the dissipative effect given by the memory.

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Mathematics Subject Classification: Primary: 35B35, 35B40, 35L51; Secondary: 35Q74, 45K05, 74D05.

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References

[1]	C. J. K. Batty, Asymptotic behaviour of semigroups of operators, in Functional Analysis and Operator Theory, Banach Center Publ. Polish Acad. Sci., Warsaw, 30 (1994), 35-52.
[2]	C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780. doi: 10.1007/s00028-008-0424-1.
[3]	C. J. K. Batty and V. Q. Phóng, Stability of individual elements under oneparameter semigroups, Trans. Amer. Math. Soc., 322 (1990), 805-818. doi: 10.1090/S0002-9947-1990-1022866-5.
[4]	A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0.
[5]	A. D. S. Campelo, D. S. Almeida Júnior and M. L. Santos, Stability of weakly dissipative Reissner-Mindlin-Timoshenko plates: A sharp result, European J. Appl. Math., 29 (2018), 226-252. doi: 10.1017/S0956792517000092.
[6]	V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymp. Anal., 46 (2006), 251-273.
[7]	I. Chueshov and I. Lasiecka, Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discret. Contin. Dyn. Syst., 15 (2006), 777-809. doi: 10.3934/dcds.2006.15.777.
[8]	I. Chueshov and I. Lasiecka, Global attractors for Mindlin-Timoshenko plates and for their Kirchhoff limits, Milan J. Math., 74 (2006), 117-138. doi: 10.1007/s00032-006-0050-8.
[9]	I. Chueshov and I. Lasiecka, On global attractors for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity, Commun. Partial Differ. Equ., 36 (2011), 67-99. doi: 10.1080/03605302.2010.484472.
[10]	M. Conti, F. Dell'Oro and V. Pata, Timoshenko systems with fading memory, Dyn. Partial Differ. Eq., 10 (2013), 367-377. doi: 10.4310/DPDE.2013.v10.n4.a4.
[11]	K.-J. Engel and R. Nagel, A short Course on Operator Semigroups, Springer, New York, 2006.
[12]	B. Feng, M. A. Jorge Silva and H. A. Caixeta, Long-time behavior for a class of semi-linear viscoelastic Kirchhoff beams/plates, Appl. Math. Optim., 82 (2020), 657-686. doi: 10.1007/s00245-018-9544-3.
[13]	H. D. Fernándes Sare, On the stability of Mindlin-Timoshenko plates, Q. Appl. Math., 67 (2009), 249-263. doi: 10.1090/S0033-569X-09-01110-2.
[14]	S. Gatti, A. Miranville, V. Pata and S. Zelik, Attractors for semi-linear equations of viscoelasticity with very low dissipation, Rocky Mount. J. Math., 38 (2008), 1117-1138. doi: 10.1216/RMJ-2008-38-4-1117.
[15]	L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394. doi: 10.1090/S0002-9947-1978-0461206-1.
[16]	C. Giorgi, M. Grasselli and V. Pata, Uniform attractors for a phase-field model with memory and quadratic nonlinearity, Indiana Univ. Math. J., 48 (1999), 1395-1445. doi: 10.1512/iumj.1999.48.1793.
[17]	C. Giorgi and M. G. Naso, Mathematical models of Reissner-Mindlin thermoviscoelastic plates, J. Therm. Stress., 29 (2006), 699-716. doi: 10.1080/01495730500499183.
[18]	C. Giorgi and F. Vegni, The longtime behavior of a nonlinear Reissner-Mindlin plate with exponentially decreasing memory kernels, J. Math. Anal. Appl., 326 (2007), 754-771. doi: 10.1016/j.jmaa.2006.03.024.
[19]	C. Giorgi and F. Vegni, Uniform energy estimates for a semilinear evolution equation of the Mindlin-Timoshenko beam with memory, Math. Comput. Model., 39 (2004), 1005-1021. doi: 10.1016/S0895-7177(04)90531-6.
[20]	M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, Prog. Nonlin. Differ. Equ. Appl., 50 (2002), 155-178. doi: 10.1007/978-3-0348-8221-7_9.
[21]	F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.
[22]	M. A. Jorge Silva, T. F. Ma and J. E. Muñoz Rivera, Mindlin-Timoshenko systems with Kelvin-Voigt: Analyticity and optimal decay rates, J. Math. Anal. Appl., 417 (2014), 164-179.
[23]	J. E. Lagnese, Boundary Stabilization of Thin Plates, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1989.
[24]	J. Lagnese and J.-L. Lions, Modelling, Analysis and Control of Thin Plates, Recherches en Mathématiques Appliquées, vol. 6. Masson, Paris, 1988.
[25]	Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC, Boca Raton, 1999.
[26]	J. E. Muñoz Rivera and H. D. Fernández Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502. doi: 10.1016/j.jmaa.2007.07.012.
[27]	V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.
[28]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
[29]	S. B. Pinheiro, Estabilidade Assintótica Para Modelos de Ondas e vigas com Derivadas de Ordens Inteiras e Fracionárias, Ph.D. thesis, State University of Maringá, Maringá, 2023.
[30]	J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.1090/S0002-9947-1984-0743749-9.
[31]	N. F. J. van Rensburg, L. Zietsman and A. J. van der Merwe, Solvability of a Reissner-Mindlin-Timoshenko plate-beam vibration model, IMA Journal of Applied Mathematics, 74 (2009), 149-162. doi: 10.1093/imamat/hxn043.