# American Institute of Mathematical Sciences

doi: 10.3934/eect.2022026
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## A general decay result for the Cauchy problem of plate equations with memory

 1 Department of Mathematics, University of Sharjah, Sharjah, United Arab Emirates, UAE 2 Laboratoire de Mathématiques Appliquées, Université Kasdi Merbah, BP 511, Ouargla, Algeria

* Corresponding author: Salim A. Messaoudi

Received  October 2021 Revised  April 2022 Early access May 2022

In this paper, we investigate the general decay rate of the solutions for a class of plate equations with memory term in the whole space
 $\mathbb{R}^n$
,
 $n\geq 1$
, given by
 $\begin{equation*} u_{tt}+\Delta^2 u+ u+ \int_0^t g(t-s)A u(s)ds = 0, \end{equation*}$
with
 $A = \Delta$
or
 $A = -Id$
. We use the energy method in the Fourier space to establish several general decay results which improve many recent results in the literature. We also present two illustrative examples by the end.
Citation: Salim A. Messaoudi, Ilyes Lacheheb. A general decay result for the Cauchy problem of plate equations with memory. Evolution Equations and Control Theory, doi: 10.3934/eect.2022026
##### References:
 [1] W. Chen and T. A. Dao, On the Cauchy problem for semilinear regularity-loss-type $\sigma$-evolution models with memory term, Nonlinear Anal. Real World Appl., 59 (2021), 103265, 26 pp. doi: 10.1016/j.nonrwa.2020.103265. [2] C. R. da Luz and R. C. Charão, Asymptotic properties for a semilinear plate equation in unbounded domains, J. Hyperbolic Differ. Equ., 6 (2009), 269-294.  doi: 10.1142/S0219891609001824. [3] I. Lacheheb and S. A. Messaoudi, General decay of the Cauchy problem for a Moore-Gibson-Thompson equation with memory, Mediterr. J. Math., 18 (2021), Paper No. 171, 21 pp. doi: 10.1007/s00009-021-01818-1. [4] J. E. Lagnese, Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping, Internat. Ser. Numer. Math., Birkhäuser-Verlag, Basel, 91 (1989), 211–236. [5] I. Lasiecka, Exponential decay rates for the solutions of Euler-Bernoulli equations with boundary dissipation occurring in the moments only, J. Differential Equations, 95 (1992), 169-182.  doi: 10.1016/0022-0396(92)90048-R. [6] I. Lasiecka, S. A. Messaoudi and M. I. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory, J. Math. Phys., 54 (2013), 031504, 18 pp. doi: 10.1063/1.4793988. [7] Y. Liu, Decay of solutions to an inertial model for a semilinear plate equation with memory, J. Math. Anal. Appl., 394 (2012), 616-632.  doi: 10.1016/j.jmaa.2012.04.003. [8] Y. Liu, Asymptotic behavior of solutions to a nonlinear plate equation with memory, Commun. Pure Appl. Anal., 16 (2017), 533-556.  doi: 10.3934/cpaa.2017027. [9] Y. Liu and S. Kawashima, Decay property for a plate equation with memory-type dissipation, Kinet. Relat. Models, 4 (2011), 531-547.  doi: 10.3934/krm.2011.4.531. [10] Y. Liu and S. Kawashima, Global existence and decay of solutions for a quasi-linear dissipative plate equation, J. Hyperbolic Differ. Equ., 8 (2011), 591-614.  doi: 10.1142/S0219891611002500. [11] Y. Liu and Y. Ueda, Decay estimate and asymptotic profile for a plate equation with memory, J. Differential Equations, 268 (2020), 2435-2463.  doi: 10.1016/j.jde.2019.09.007. [12] S. Mao and Y. Liu, Decay of solutions to generalized plate type equations with memory, Kinet. Relat. Models, 7 (2014), 121-131.  doi: 10.3934/krm.2014.7.121. [13] J. E. Muñoz Rivera, E. C. Lapa and R. Barreto, Decay rates for viscoelastic plates with memory, J. Elasticity, 44 (1996), 61-87.  doi: 10.1007/BF00042192. [14] J. E. Muñoz Rivera, M. G. Naso and F. M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl., 286 (2003), 692-704.  doi: 10.1016/S0022-247X(03)00511-0. [15] J. E. Muñoz Rivera and Y. Shibata, A linear thermoelastic plate equation with Dirichlet boundary condition, Math. Methods Appl. Sci., 20 (1997), 915-932.  doi: 10.1002/(SICI)1099-1476(19970725)20:11<915::AID-MMA891>3.0.CO;2-4. [16] M. I. Mustafa, General decay result for nonlinear viscoelastic equations, J. Math. Anal. Appl., 457 (2018), 134-152.  doi: 10.1016/j.jmaa.2017.08.019. [17] M. I. Mustafa and S. A. Messaoudi, General stability result for viscoelastic wave equations, J. Math. Phys., 53 (2012), 053702, 14 pp. doi: 10.1063/1.4711830. [18] B. Said-Houari and S. A. Messaoudi, General decay estimates for a Cauchy viscoelastic wave problem, Commun. Pure Appl. Anal., 13 (2014), 1541-1551.  doi: 10.3934/cpaa.2014.13.1541. [19] Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semilinear dissipative plate equation, J. Hyperbolic Differ. Equ., 7 (2010), 471-501.  doi: 10.1142/S0219891610002207. [20] R. Temam, Navier-Stokes equations, revised edition, Studies in Mathematics and its Applications, vol. 2, North-Holland, Amsterdam, New York, Oxford, 1979. [21] J. Wirth, Asymptotic Properties of Solutions to Wave Equations with Time-Dependent Dissipation, Ph.D thesis, TU Bergakademie Freiberg, 2004.

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##### References:
 [1] W. Chen and T. A. Dao, On the Cauchy problem for semilinear regularity-loss-type $\sigma$-evolution models with memory term, Nonlinear Anal. Real World Appl., 59 (2021), 103265, 26 pp. doi: 10.1016/j.nonrwa.2020.103265. [2] C. R. da Luz and R. C. Charão, Asymptotic properties for a semilinear plate equation in unbounded domains, J. Hyperbolic Differ. Equ., 6 (2009), 269-294.  doi: 10.1142/S0219891609001824. [3] I. Lacheheb and S. A. Messaoudi, General decay of the Cauchy problem for a Moore-Gibson-Thompson equation with memory, Mediterr. J. Math., 18 (2021), Paper No. 171, 21 pp. doi: 10.1007/s00009-021-01818-1. [4] J. E. Lagnese, Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping, Internat. Ser. Numer. Math., Birkhäuser-Verlag, Basel, 91 (1989), 211–236. [5] I. Lasiecka, Exponential decay rates for the solutions of Euler-Bernoulli equations with boundary dissipation occurring in the moments only, J. Differential Equations, 95 (1992), 169-182.  doi: 10.1016/0022-0396(92)90048-R. [6] I. Lasiecka, S. A. Messaoudi and M. I. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory, J. Math. Phys., 54 (2013), 031504, 18 pp. doi: 10.1063/1.4793988. [7] Y. Liu, Decay of solutions to an inertial model for a semilinear plate equation with memory, J. Math. Anal. Appl., 394 (2012), 616-632.  doi: 10.1016/j.jmaa.2012.04.003. [8] Y. Liu, Asymptotic behavior of solutions to a nonlinear plate equation with memory, Commun. Pure Appl. Anal., 16 (2017), 533-556.  doi: 10.3934/cpaa.2017027. [9] Y. Liu and S. Kawashima, Decay property for a plate equation with memory-type dissipation, Kinet. Relat. Models, 4 (2011), 531-547.  doi: 10.3934/krm.2011.4.531. [10] Y. Liu and S. Kawashima, Global existence and decay of solutions for a quasi-linear dissipative plate equation, J. Hyperbolic Differ. Equ., 8 (2011), 591-614.  doi: 10.1142/S0219891611002500. [11] Y. Liu and Y. Ueda, Decay estimate and asymptotic profile for a plate equation with memory, J. Differential Equations, 268 (2020), 2435-2463.  doi: 10.1016/j.jde.2019.09.007. [12] S. Mao and Y. Liu, Decay of solutions to generalized plate type equations with memory, Kinet. Relat. Models, 7 (2014), 121-131.  doi: 10.3934/krm.2014.7.121. [13] J. E. Muñoz Rivera, E. C. Lapa and R. Barreto, Decay rates for viscoelastic plates with memory, J. Elasticity, 44 (1996), 61-87.  doi: 10.1007/BF00042192. [14] J. E. Muñoz Rivera, M. G. Naso and F. M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl., 286 (2003), 692-704.  doi: 10.1016/S0022-247X(03)00511-0. [15] J. E. Muñoz Rivera and Y. Shibata, A linear thermoelastic plate equation with Dirichlet boundary condition, Math. Methods Appl. Sci., 20 (1997), 915-932.  doi: 10.1002/(SICI)1099-1476(19970725)20:11<915::AID-MMA891>3.0.CO;2-4. [16] M. I. Mustafa, General decay result for nonlinear viscoelastic equations, J. Math. Anal. Appl., 457 (2018), 134-152.  doi: 10.1016/j.jmaa.2017.08.019. [17] M. I. Mustafa and S. A. Messaoudi, General stability result for viscoelastic wave equations, J. Math. Phys., 53 (2012), 053702, 14 pp. doi: 10.1063/1.4711830. [18] B. Said-Houari and S. A. Messaoudi, General decay estimates for a Cauchy viscoelastic wave problem, Commun. Pure Appl. Anal., 13 (2014), 1541-1551.  doi: 10.3934/cpaa.2014.13.1541. [19] Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semilinear dissipative plate equation, J. Hyperbolic Differ. Equ., 7 (2010), 471-501.  doi: 10.1142/S0219891610002207. [20] R. Temam, Navier-Stokes equations, revised edition, Studies in Mathematics and its Applications, vol. 2, North-Holland, Amsterdam, New York, Oxford, 1979. [21] J. Wirth, Asymptotic Properties of Solutions to Wave Equations with Time-Dependent Dissipation, Ph.D thesis, TU Bergakademie Freiberg, 2004.
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