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July  2019, 39(7): 3839-3866. doi: 10.3934/dcds.2019155

General decay of the solution to a nonlinear viscoelastic modified von-Kármán system with delay

Faculty of Mathematics, University of Science and Technology Houari Boumedienne, P.O. Box 32, El-Alia 16111, Bab Ezzouar, Algiers, Algeria

* Corresponding author: Ammar Khemmoudj

Received  April 2018 Revised  January 2019 Published  April 2019

In this paper we consider a viscoelastic modified nonlinear Von-Kármán system with a linear delay term. The well posedness of solutions is proved using the Faedo-Galerkin method. We use minimal and general conditions on the relaxation function and establish a general decay results, from which the usual exponential and polynomial decay rates are only special cases.

Citation: Ammar Khemmoudj, Yacine Mokhtari. General decay of the solution to a nonlinear viscoelastic modified von-Kármán system with delay. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3839-3866. doi: 10.3934/dcds.2019155
References:
[1]

C. Abdallah, P. Dorato, J. Benitez-Read and R. Byrne, Delayed positive feedback can stabilize oscillatory systems, Proceedings of the American Control Conference, 2-4 June, San Francisco, CA, (1993), 3106–3107. doi: 10.23919/ACC.1993.4793475. Google Scholar

[2]

F. Alabau-Boussouira, On convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Applied Mathematics and Optimization, 51 (2005), 61-105. doi: 10.1007/s00245. Google Scholar

[3]

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

[4]

F. D. ArarunaP. B. E Silva and E. Zuazua, Asymptotic limits and stabilization for the 1-D Nonlinear Mindlin-Timoshenko system, J. Syst. Sci. Complex., 23 (2010), 1-17. doi: 10.1007/s11424-010-0137-8. Google Scholar

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V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1. Google Scholar

[6]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5. Google Scholar

[7]

A. Benaissa and A. Guesmia, Energy decay for wave equations of $\varphi$-Laplacian type with weakly nonlinear dissipation, Electronic Journal of Differential Equations, 2008, 1–22. Google Scholar

[8]

A. Benaissa and N. Louhibi, Global existence and Energy decay of solutions to a nonlinear wave equation with a delay term, Georgian Mathematical Journal, 20 (2013), 1-24. doi: 10.1515/gmj-2013-0006. Google Scholar

[9]

A. BenaissaA. Benguessoum and S. A. Messaoudi, Global existence and energy decay of solutions to a viscoelastic wave equation with a delay term in the nonlinear internal feedback, Int. J. Dynamical Systems and Differential Equations, 5 (2014), 1-26. doi: 10.1504/IJDSDE.2014.067080. Google Scholar

[10]

S. Berrimi and S. A. Messaoudi, Existence and decay of solutions of a viscoelastic equation with a nonlinear source, Nonlinear Analysis: Theory, Methods and Applications, 64 (2006), 2314-2331. doi: 10.1016/j.na.2005.08.015. Google Scholar

[11]

H. Brezis, Analyse Fonctionnelle: Theéorie et Applications, Dunod, Paris, 1983. Google Scholar

[12]

M. M. CavalcantiV. D. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, Journal of Differential Equations, 236 (2007), 407-459. doi: 10.1016/j.jde.2007.02.004. Google Scholar

[13]

M. M. CavalcantiV. N. Domingos Cavalcanti and A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electronic Journal of Differential Equations, (2002), 1-14. doi: 10.1016/j.na.2006.10.040. Google Scholar

[14]

M. M. CavalcantiV. N. Domingos Cavalcanti and P. Martinez, General Decay Rate Estimates for Viscoelastic Dissipative Systems, Nonlinear Analysis, 68 (2008), 177-193. doi: 10.1016/j.na.2006.10.040. Google Scholar

[15]

M. M. CavalcantiV. N. Domingos Cavalcanti and M. L. Santos, Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary, Appl. Math. Comput., 150 (2004), 439-465. doi: 10.1016/S0096-3003(03)00284-4. Google Scholar

[16]

I. ChueshovM. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differ. Equ., 27 (2002), 1901-1951. doi: 10.1081/PDE-120016132. Google Scholar

[17]

I. Chueshov and I. Lasiecka, Global Attractors for von Kármán Evolutions with a Nonlinear Boundary Dissipation, Journal of Differential Equations, 198 (2004), 196-231. doi: 10.1016/j.jde.2003.08.008. Google Scholar

[18]

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. Google Scholar

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C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. doi: 10.1007/BF00251609. Google Scholar

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M. DaoulatliI. Lasiecka and D. Toundykov, Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions, Discrete and Continuous Dynamical Systems, 2 (2009), 67-94. doi: 10.3934/dcdss.2009.2.67. Google Scholar

[21]

R. DatkoJ. E. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM Journal on Control and Optimization, 24 (1986), 152-156. doi: 10.1137/0324007. Google Scholar

[22]

R. Datko, Not All Feedback Stabilized Hyperbolic Systems Are Robust with Respect to Small Time Delays in Their Feedbacks, SIAM Journal on Control and Optimization, 26 (1988), 697-713. doi: 10.1137/0326040. Google Scholar

[23]

J. F. Doyle, Wave Propagation in Structures, Springer-Verlag, New York, 1997.Google Scholar

[24]

M. EllerJ. E. Lagnese and S. Nicaise, Decay rates for solutions of a Maxwell system with nonlinear boundary damping, Computational and Applied Mathematics, 21 (2002), 135-165. Google Scholar

[25]

A. FaviniM. A. HornI. Lasiecka and D. Tartaru, Global existence, uniqueness and regularity of solutions to a von Kármán system with nonlinear boundary dissipation, Diff. Integ. Eqns, 9 (1996), 267-294. Google Scholar

[26]

X. Han and M. Wang, General decay of energy for a viscoelastic equation with nonlinear damping, Math. Methods Appl. Sci., 32 (2009), 346-358. doi: 10.1002/mma.1041. Google Scholar

[27] G. H. HardyJ. E. Littlewood and G. Polya, Inequalities,, Cambridge University Press, Cambridge, 1988. Google Scholar
[28]

M. KafiniS. A. Messaoudi and M. I. Mustafa, Energy decay rates for a timoshenko-type system of thermoelasticity of type III with constant delay, Applicable Analysis, 93 (2014), 1201-1216. doi: 10.1080/00036811.2013.823480. Google Scholar

[29]

V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, Masson-John Wiley, Paris, 1994. Google Scholar

[30]

J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM, 1989. doi: 10.1137/1.9781611970821. Google Scholar

[31]

J. E. Lagnese and G. Leucering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback, Journal of differential equation, 91 (1991), 355-388. doi: 10.1016/0022-0396(91)90145-Y. Google Scholar

[32]

J. E. Lagnese and J. L. Lions, Modelling Analysis and Control of Thin Plates, RMA 6, Masson, Paris, 1988. Google Scholar

[33]

J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia, PA., 1989. doi: 10.1137/1.9781611970821. Google Scholar

[34]

I. Lasiecka, Mathematical Control Theory of Coupled PDE's (CBMS-NSF Regional Conference Series in Applied Mathematics), SIAM, Philadelphia, PA, 2002. doi: 10.1137/1.9780898717099. Google Scholar

[35]

I. Lasiecka and D. Doundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and a nonlinear source, Nonlinear Analysis, 64 (2006), 1757-1797. doi: 10.1016/j.na.2005.07.024. Google Scholar

[36]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential and Integral Equations, 6 (1993), 507-533. Google Scholar

[37]

I. Lasiecka and X. Wang, Intrinsic decay rate estimates for semilinear abstract second order equations with memory, in: New Prospects in Direct, Inverse and Control Problems for Evolution Equations, in: Springer INdAM, Springer, Cham, 10 (2014), 271–303. doi: 10.1007/978-3-319-11406-4_14. Google Scholar

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M. J. LeeJ. Y. Park and Y. H. Kang, Exponential decay rate for a quasilinear von Kármán equation of memory type with acoustic boundary conditions, Boundary Value Problems, 2015 (2015), 14pp. doi: 10.1186/s13661-015-0381-x. Google Scholar

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J. L. Lions, Quelques Methodes de Resolution des Problemes Aux Limites non Lineaires, Dunod, Paris, (in French) 1969. Google Scholar

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W. J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75. Google Scholar

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S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Analysis, 69 (2008), 2589-2598. doi: 10.1016/j.na.2007.08.035. Google Scholar

[42]

S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467. doi: 10.1016/j.jmaa.2007.11.048. Google Scholar

[43]

J. E. Munoz Rivera and G. P. Menzala, Decay rates of solutions of a von Kármán system for viscoelastic plates with memory, Quarterly of Applied Mathematics, 57 (1999), 181-200. doi: 10.1090/qam/1672191. Google Scholar

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J. E. Munoz RiveraH. Portillo Oquendo and M. L. Santos, Asymptotic behavior to a von Kármán plate with boundary memory conditions, Nonlinear Analysis, 62 (2005), 1183-1205. doi: 10.1016/j.na.2005.04.025. Google Scholar

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M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Math. Meth. Appl. Sci., 41 (2018), 192-204. doi: 10.1002/mma.4604. Google Scholar

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M. I. Mustafa, Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations, Nonlinear Anal., Real World Appl., 13 (2012), 452-463. doi: 10.1016/j.nonrwa.2011.08.002. Google Scholar

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M. I. Mustafa, Uniform decay rates for viscoelastic dissipative systems, J. Dyn. Control Syst., 22 (2016), 101-116. doi: 10.1007/s10883-014-9256-1. Google Scholar

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M. I. Mustapha, General decay result for nonlinear viscoelastic equation, J. Math. Anal. Appl., 457 (2018), 134-152. doi: 10.1016/j.jmaa.2017.08.019. Google Scholar

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M. I. Mustapha, Laminated Timoshenko beams with viscoelastic damping, J. Math. Anal. Appl., 466 (2018), 619-641. doi: 10.1016/j.jmaa.2018.06.016. Google Scholar

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M. I. Mustapha and S. A. Messaoudi, General stability result for viscoelastic wave equations, J. Math. Phys., 53 (2012), 053702, 14pp. doi: 10.1063/1.4711830. Google Scholar

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S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM Journal on Control and Optimization, 45 (2006), 1561-1585. doi: 10.1137/060648891. Google Scholar

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S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differ. Integral Equ., 21 (2008), 935-958. Google Scholar

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J. Y. Park and S. H. Park, Uniform Decay for a von Kármán Plate Equation with a Boundary Memory Condition, Mathematical Methods in the Applied Sciences, 28 (2005), 2225-2240. doi: 10.1002/mma.663. Google Scholar

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S. H. ParkJ. Y. Park and Y. H. Kang, General Decay for a von Kármán equation of memory type with acoustic boundary conditions, Zeitschrift für Angewandte Mathematik und Physik, 63 (2012), 813-823. doi: 10.1007/s00033-011-0188-2. Google Scholar

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show all references

References:
[1]

C. Abdallah, P. Dorato, J. Benitez-Read and R. Byrne, Delayed positive feedback can stabilize oscillatory systems, Proceedings of the American Control Conference, 2-4 June, San Francisco, CA, (1993), 3106–3107. doi: 10.23919/ACC.1993.4793475. Google Scholar

[2]

F. Alabau-Boussouira, On convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Applied Mathematics and Optimization, 51 (2005), 61-105. doi: 10.1007/s00245. Google Scholar

[3]

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

[4]

F. D. ArarunaP. B. E Silva and E. Zuazua, Asymptotic limits and stabilization for the 1-D Nonlinear Mindlin-Timoshenko system, J. Syst. Sci. Complex., 23 (2010), 1-17. doi: 10.1007/s11424-010-0137-8. Google Scholar

[5]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1. Google Scholar

[6]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5. Google Scholar

[7]

A. Benaissa and A. Guesmia, Energy decay for wave equations of $\varphi$-Laplacian type with weakly nonlinear dissipation, Electronic Journal of Differential Equations, 2008, 1–22. Google Scholar

[8]

A. Benaissa and N. Louhibi, Global existence and Energy decay of solutions to a nonlinear wave equation with a delay term, Georgian Mathematical Journal, 20 (2013), 1-24. doi: 10.1515/gmj-2013-0006. Google Scholar

[9]

A. BenaissaA. Benguessoum and S. A. Messaoudi, Global existence and energy decay of solutions to a viscoelastic wave equation with a delay term in the nonlinear internal feedback, Int. J. Dynamical Systems and Differential Equations, 5 (2014), 1-26. doi: 10.1504/IJDSDE.2014.067080. Google Scholar

[10]

S. Berrimi and S. A. Messaoudi, Existence and decay of solutions of a viscoelastic equation with a nonlinear source, Nonlinear Analysis: Theory, Methods and Applications, 64 (2006), 2314-2331. doi: 10.1016/j.na.2005.08.015. Google Scholar

[11]

H. Brezis, Analyse Fonctionnelle: Theéorie et Applications, Dunod, Paris, 1983. Google Scholar

[12]

M. M. CavalcantiV. D. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, Journal of Differential Equations, 236 (2007), 407-459. doi: 10.1016/j.jde.2007.02.004. Google Scholar

[13]

M. M. CavalcantiV. N. Domingos Cavalcanti and A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electronic Journal of Differential Equations, (2002), 1-14. doi: 10.1016/j.na.2006.10.040. Google Scholar

[14]

M. M. CavalcantiV. N. Domingos Cavalcanti and P. Martinez, General Decay Rate Estimates for Viscoelastic Dissipative Systems, Nonlinear Analysis, 68 (2008), 177-193. doi: 10.1016/j.na.2006.10.040. Google Scholar

[15]

M. M. CavalcantiV. N. Domingos Cavalcanti and M. L. Santos, Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary, Appl. Math. Comput., 150 (2004), 439-465. doi: 10.1016/S0096-3003(03)00284-4. Google Scholar

[16]

I. ChueshovM. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differ. Equ., 27 (2002), 1901-1951. doi: 10.1081/PDE-120016132. Google Scholar

[17]

I. Chueshov and I. Lasiecka, Global Attractors for von Kármán Evolutions with a Nonlinear Boundary Dissipation, Journal of Differential Equations, 198 (2004), 196-231. doi: 10.1016/j.jde.2003.08.008. Google Scholar

[18]

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. Google Scholar

[19]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. doi: 10.1007/BF00251609. Google Scholar

[20]

M. DaoulatliI. Lasiecka and D. Toundykov, Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions, Discrete and Continuous Dynamical Systems, 2 (2009), 67-94. doi: 10.3934/dcdss.2009.2.67. Google Scholar

[21]

R. DatkoJ. E. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM Journal on Control and Optimization, 24 (1986), 152-156. doi: 10.1137/0324007. Google Scholar

[22]

R. Datko, Not All Feedback Stabilized Hyperbolic Systems Are Robust with Respect to Small Time Delays in Their Feedbacks, SIAM Journal on Control and Optimization, 26 (1988), 697-713. doi: 10.1137/0326040. Google Scholar

[23]

J. F. Doyle, Wave Propagation in Structures, Springer-Verlag, New York, 1997.Google Scholar

[24]

M. EllerJ. E. Lagnese and S. Nicaise, Decay rates for solutions of a Maxwell system with nonlinear boundary damping, Computational and Applied Mathematics, 21 (2002), 135-165. Google Scholar

[25]

A. FaviniM. A. HornI. Lasiecka and D. Tartaru, Global existence, uniqueness and regularity of solutions to a von Kármán system with nonlinear boundary dissipation, Diff. Integ. Eqns, 9 (1996), 267-294. Google Scholar

[26]

X. Han and M. Wang, General decay of energy for a viscoelastic equation with nonlinear damping, Math. Methods Appl. Sci., 32 (2009), 346-358. doi: 10.1002/mma.1041. Google Scholar

[27] G. H. HardyJ. E. Littlewood and G. Polya, Inequalities,, Cambridge University Press, Cambridge, 1988. Google Scholar
[28]

M. KafiniS. A. Messaoudi and M. I. Mustafa, Energy decay rates for a timoshenko-type system of thermoelasticity of type III with constant delay, Applicable Analysis, 93 (2014), 1201-1216. doi: 10.1080/00036811.2013.823480. Google Scholar

[29]

V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, Masson-John Wiley, Paris, 1994. Google Scholar

[30]

J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM, 1989. doi: 10.1137/1.9781611970821. Google Scholar

[31]

J. E. Lagnese and G. Leucering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback, Journal of differential equation, 91 (1991), 355-388. doi: 10.1016/0022-0396(91)90145-Y. Google Scholar

[32]

J. E. Lagnese and J. L. Lions, Modelling Analysis and Control of Thin Plates, RMA 6, Masson, Paris, 1988. Google Scholar

[33]

J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia, PA., 1989. doi: 10.1137/1.9781611970821. Google Scholar

[34]

I. Lasiecka, Mathematical Control Theory of Coupled PDE's (CBMS-NSF Regional Conference Series in Applied Mathematics), SIAM, Philadelphia, PA, 2002. doi: 10.1137/1.9780898717099. Google Scholar

[35]

I. Lasiecka and D. Doundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and a nonlinear source, Nonlinear Analysis, 64 (2006), 1757-1797. doi: 10.1016/j.na.2005.07.024. Google Scholar

[36]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential and Integral Equations, 6 (1993), 507-533. Google Scholar

[37]

I. Lasiecka and X. Wang, Intrinsic decay rate estimates for semilinear abstract second order equations with memory, in: New Prospects in Direct, Inverse and Control Problems for Evolution Equations, in: Springer INdAM, Springer, Cham, 10 (2014), 271–303. doi: 10.1007/978-3-319-11406-4_14. Google Scholar

[38]

M. J. LeeJ. Y. Park and Y. H. Kang, Exponential decay rate for a quasilinear von Kármán equation of memory type with acoustic boundary conditions, Boundary Value Problems, 2015 (2015), 14pp. doi: 10.1186/s13661-015-0381-x. Google Scholar

[39]

J. L. Lions, Quelques Methodes de Resolution des Problemes Aux Limites non Lineaires, Dunod, Paris, (in French) 1969. Google Scholar

[40]

W. J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75. Google Scholar

[41]

S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Analysis, 69 (2008), 2589-2598. doi: 10.1016/j.na.2007.08.035. Google Scholar

[42]

S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467. doi: 10.1016/j.jmaa.2007.11.048. Google Scholar

[43]

J. E. Munoz Rivera and G. P. Menzala, Decay rates of solutions of a von Kármán system for viscoelastic plates with memory, Quarterly of Applied Mathematics, 57 (1999), 181-200. doi: 10.1090/qam/1672191. Google Scholar

[44]

J. E. Munoz RiveraH. Portillo Oquendo and M. L. Santos, Asymptotic behavior to a von Kármán plate with boundary memory conditions, Nonlinear Analysis, 62 (2005), 1183-1205. doi: 10.1016/j.na.2005.04.025. Google Scholar

[45]

M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Math. Meth. Appl. Sci., 41 (2018), 192-204. doi: 10.1002/mma.4604. Google Scholar

[46]

M. I. Mustafa, Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations, Nonlinear Anal., Real World Appl., 13 (2012), 452-463. doi: 10.1016/j.nonrwa.2011.08.002. Google Scholar

[47]

M. I. Mustafa, Uniform decay rates for viscoelastic dissipative systems, J. Dyn. Control Syst., 22 (2016), 101-116. doi: 10.1007/s10883-014-9256-1. Google Scholar

[48]

M. I. Mustapha, General decay result for nonlinear viscoelastic equation, J. Math. Anal. Appl., 457 (2018), 134-152. doi: 10.1016/j.jmaa.2017.08.019. Google Scholar

[49]

M. I. Mustapha, Laminated Timoshenko beams with viscoelastic damping, J. Math. Anal. Appl., 466 (2018), 619-641. doi: 10.1016/j.jmaa.2018.06.016. Google Scholar

[50]

M. I. Mustapha and S. A. Messaoudi, General stability result for viscoelastic wave equations, J. Math. Phys., 53 (2012), 053702, 14pp. doi: 10.1063/1.4711830. Google Scholar

[51]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM Journal on Control and Optimization, 45 (2006), 1561-1585. doi: 10.1137/060648891. Google Scholar

[52]

S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differ. Integral Equ., 21 (2008), 935-958. Google Scholar

[53]

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