March  2020, 28(1): 311-326. doi: 10.3934/era.2020018

Long-time behavior of a class of viscoelastic plate equations

College of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730124, China; College of Mathematics, Sichuan University, Chengdu 610065, China

* Corresponding author: Yang Liu

Received  December 2019 Published  March 2020

Fund Project: The author is supported by the Fundamental Research Funds for the Central Universities (Grant No. 31920190090)

This paper is concerned with the initial-boundary value problem for a class of viscoelastic plate equations on an arbitrary dimensional bounded domain. Under certain assumptions on the memory kernel and the source term, the global well-posedness of solutions and the existence of global attractors are obtained.

Citation: Yang Liu. Long-time behavior of a class of viscoelastic plate equations. Electronic Research Archive, 2020, 28 (1) : 311-326. doi: 10.3934/era.2020018
References:
[1]

F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Math. Acad. Sci. Paris, 347 (2009), 867-872.  doi: 10.1016/j.crma.2009.05.011.  Google Scholar

[2]

F. Alabau-BoussouiraP. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254 (2008), 1342-1372.  doi: 10.1016/j.jfa.2007.09.012.  Google Scholar

[3]

S. Borini and V. Pata, Uniform attractors for a strongly damped wave equation with linear memory, Asymptot. Anal., 20 (1999), 263-277.   Google Scholar

[4]

P. Cannarsa and D. Sforza, Integro-differential equations of hyperbolic type with positive definite kernels, J. Differential Equations, 250 (2011), 4289-4335.  doi: 10.1016/j.jde.2011.03.005.  Google Scholar

[5]

M. M. CavalcantiV. N. D. CavalcantiI. Lasiecka and F. A. F. Nascimento, Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1987-2012.  doi: 10.3934/dcdsb.2014.19.1987.  Google Scholar

[6]

M. M. CavalcantiV. N. D. Cavalcanti and T. F. Ma, Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains, Differential Integral Equations, 17 (2004), 495-510.   Google Scholar

[7]

V. V. Chepyzhov and A. Miranville, Trajectory and global attractors of dissipative hyperbolic equations with memory, Commun. Pure Appl. Anal., 4 (2005), 115-142.  doi: 10.3934/cpaa.2005.4.115.  Google Scholar

[8]

I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc., Rhode Island, 2008. doi: 10.1090/memo/0912.  Google Scholar

[9]

M. Conti and P. G. Geredeli, Existence of smooth global attractors for nonlinear viscoelastic equations with memory, J. Evol. Equ., 15 (2015), 533-558.  doi: 10.1007/s00028-014-0270-2.  Google Scholar

[10]

J.-M. Ghidaglia and A. Marzocchi, Longtime behaviour of strongly damped wave equations, global attractors and their dimension, SIAM J. Math. Anal., 22 (1991), 879-895.  doi: 10.1137/0522057.  Google Scholar

[11]

C. GiorgiV. Pata and A. Marzocchi, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 333-354.  doi: 10.1007/s000300050049.  Google Scholar

[12]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in Evolution Equations, Semigroups and Functional Analysis, Birkhäuser, Basel, 2002,155â€"178.  Google Scholar

[13]

A. Guesmia and S. A. Messaoudi, A new approach to the stability of an abstract system in the presence of infinite history, J. Math. Anal. Appl., 416 (2014), 212-228.  doi: 10.1016/j.jmaa.2014.02.030.  Google Scholar

[14]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[15]

M. A. J. Silva and 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.  doi: 10.1093/imamat/hxs011.  Google Scholar

[16] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569418.  Google Scholar
[17]

J. E. Lagnese, Boundary Stabilization of Thin Plates, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.  Google Scholar

[18]

I. Lasiecka and X. Wang, Intrinsic decay rate estimates for semilinear abstract second order equations with memory, New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer, Cham, 2014,271â€"303. doi: 10.1007/978-3-319-11406-4_14.  Google Scholar

[19]

A. Marzocchi and E. Vuk, Global attractor for damped semilinear elastic beam equations with memory, Z. Angew. Math. Phys., 54 (2003), 224-234.  doi: 10.1007/s000330300002.  Google Scholar

[20]

J. E. M. Rivera and L. H. Fatori, Smoothing effect and propagations of singularities for viscoelastic plates, J. Math. Anal. Appl., 206 (1997), 397-427.  doi: 10.1006/jmaa.1997.5223.  Google Scholar

[21]

J. E. M. RiveraE. C. Lapa and R. Barreto, Decay rates for viscoelastic plates with memory, J. Elasticity, 44 (1996), 61-87.  doi: 10.1007/BF00042192.  Google Scholar

[22]

J. E. M. Rivera and M. G. Naso, Asymptotic stability of semigroups associated with linear weak dissipative systems with memory, J. Math. Anal. Appl., 326 (2007), 691-707.  doi: 10.1016/j.jmaa.2006.03.022.  Google Scholar

[23]

V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math., 64 (2006), 499-513.  doi: 10.1090/S0033-569X-06-01010-4.  Google Scholar

[24]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar

[25]

J. Prüss, Decay properties for the solutions of a partial differential equation with memory, Arch. Math., 92 (2009), 158-173.  doi: 10.1007/s00013-008-2936-x.  Google Scholar

[26]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[27]

Z. Yang and B. Jin, Global attractor for a class of Kirchhoff models, J. Math. Phys., 50 (2009), 29 pp. doi: 10.1063/1.3085951.  Google Scholar

show all references

References:
[1]

F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Math. Acad. Sci. Paris, 347 (2009), 867-872.  doi: 10.1016/j.crma.2009.05.011.  Google Scholar

[2]

F. Alabau-BoussouiraP. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254 (2008), 1342-1372.  doi: 10.1016/j.jfa.2007.09.012.  Google Scholar

[3]

S. Borini and V. Pata, Uniform attractors for a strongly damped wave equation with linear memory, Asymptot. Anal., 20 (1999), 263-277.   Google Scholar

[4]

P. Cannarsa and D. Sforza, Integro-differential equations of hyperbolic type with positive definite kernels, J. Differential Equations, 250 (2011), 4289-4335.  doi: 10.1016/j.jde.2011.03.005.  Google Scholar

[5]

M. M. CavalcantiV. N. D. CavalcantiI. Lasiecka and F. A. F. Nascimento, Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1987-2012.  doi: 10.3934/dcdsb.2014.19.1987.  Google Scholar

[6]

M. M. CavalcantiV. N. D. Cavalcanti and T. F. Ma, Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains, Differential Integral Equations, 17 (2004), 495-510.   Google Scholar

[7]

V. V. Chepyzhov and A. Miranville, Trajectory and global attractors of dissipative hyperbolic equations with memory, Commun. Pure Appl. Anal., 4 (2005), 115-142.  doi: 10.3934/cpaa.2005.4.115.  Google Scholar

[8]

I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc., Rhode Island, 2008. doi: 10.1090/memo/0912.  Google Scholar

[9]

M. Conti and P. G. Geredeli, Existence of smooth global attractors for nonlinear viscoelastic equations with memory, J. Evol. Equ., 15 (2015), 533-558.  doi: 10.1007/s00028-014-0270-2.  Google Scholar

[10]

J.-M. Ghidaglia and A. Marzocchi, Longtime behaviour of strongly damped wave equations, global attractors and their dimension, SIAM J. Math. Anal., 22 (1991), 879-895.  doi: 10.1137/0522057.  Google Scholar

[11]

C. GiorgiV. Pata and A. Marzocchi, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 333-354.  doi: 10.1007/s000300050049.  Google Scholar

[12]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in Evolution Equations, Semigroups and Functional Analysis, Birkhäuser, Basel, 2002,155â€"178.  Google Scholar

[13]

A. Guesmia and S. A. Messaoudi, A new approach to the stability of an abstract system in the presence of infinite history, J. Math. Anal. Appl., 416 (2014), 212-228.  doi: 10.1016/j.jmaa.2014.02.030.  Google Scholar

[14]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[15]

M. A. J. Silva and 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.  doi: 10.1093/imamat/hxs011.  Google Scholar

[16] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569418.  Google Scholar
[17]

J. E. Lagnese, Boundary Stabilization of Thin Plates, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.  Google Scholar

[18]

I. Lasiecka and X. Wang, Intrinsic decay rate estimates for semilinear abstract second order equations with memory, New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer, Cham, 2014,271â€"303. doi: 10.1007/978-3-319-11406-4_14.  Google Scholar

[19]

A. Marzocchi and E. Vuk, Global attractor for damped semilinear elastic beam equations with memory, Z. Angew. Math. Phys., 54 (2003), 224-234.  doi: 10.1007/s000330300002.  Google Scholar

[20]

J. E. M. Rivera and L. H. Fatori, Smoothing effect and propagations of singularities for viscoelastic plates, J. Math. Anal. Appl., 206 (1997), 397-427.  doi: 10.1006/jmaa.1997.5223.  Google Scholar

[21]

J. E. M. RiveraE. C. Lapa and R. Barreto, Decay rates for viscoelastic plates with memory, J. Elasticity, 44 (1996), 61-87.  doi: 10.1007/BF00042192.  Google Scholar

[22]

J. E. M. Rivera and M. G. Naso, Asymptotic stability of semigroups associated with linear weak dissipative systems with memory, J. Math. Anal. Appl., 326 (2007), 691-707.  doi: 10.1016/j.jmaa.2006.03.022.  Google Scholar

[23]

V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math., 64 (2006), 499-513.  doi: 10.1090/S0033-569X-06-01010-4.  Google Scholar

[24]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar

[25]

J. Prüss, Decay properties for the solutions of a partial differential equation with memory, Arch. Math., 92 (2009), 158-173.  doi: 10.1007/s00013-008-2936-x.  Google Scholar

[26]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[27]

Z. Yang and B. Jin, Global attractor for a class of Kirchhoff models, J. Math. Phys., 50 (2009), 29 pp. doi: 10.1063/1.3085951.  Google Scholar

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