Decay rates for a coupled viscoelastic Lamé system with strong damping
Abstract
In [6] Beniani, Taouaf and Benaissa studied a coupled viscoelastic Lamé system with strong dampings and established a general decay result. In this paper, we continue to study the system. Assuming gi0(t) ≤−ξi(t)Hi(gi(t)), i = 1,2, we establish an explicit and general decay result, which is optimal, to the system. This result improves earlier results in [6].
Keyword : Lamé system, energy decay, viscoelastic damping, convexity
How to Cite
Feng, B., & Li, H. (2020). Decay rates for a coupled viscoelastic Lamé system with strong damping. Mathematical Modelling and Analysis, 25(2), 226-240. https://doi.org/10.3846/mma.2020.10383
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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M.M. Cavalcanti, V.N.D. Cavalcanti, I. Lasiecka and X. Wang. Existence and sharp decay rate estimates for a von Karman system with long memory. Nonlinear Anal.: Real World Appl., 22:289–306, 2015. https://doi.org/10.1016/j.nonrwa.2014.09.016
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I. Lasiecka, S.A. Messaoudi and M.I. Mustafa. Note on intrinsic decay rates for abstract wave equations with memory. J. Math. Phys., 54(3), 2013. https://doi.org/10.1063/1.4793988
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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, Springer INdAM Series, 10:271–303, 2014.
W.J. Liu. General decay of solutions to a viscoelastic wave equation with nonlinear localized damping. Ann. Acad. Sci. Fenn. Math., 34:291–302, 2009.
W.J. Liu. General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms. J. Math. Phys., 50:113506, 2009. https://doi.org/10.1063/1.3254323
S.A. Messaoud and M.I. Mustafa. On the control of solutions of viscoelastic equations with boundary feedback. Nonlinear Anal.: Real World Appl., 10(5):3132– 3140, 2009. https://doi.org/10.1016/j.nonrwa.2008.10.026
S.A. Messaoudi. General decay of solutions of a viscoelastic equation. J. Math. Anal. Appl., 341(2):1457–1467, 2008. https://doi.org/10.1016/j.jmaa.2007.11.048
S.A. Messaoudi. General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Anal., 69(8):2589–2598, 2008. https://doi.org/10.1016/j.na.2007.08.035
S.A. Messaoudi and M.M. Al-Gharabli. A general decay result of a nonlinear system of wave equations with infinite memories. Appl. Math. Compu., 259(15):540– 551, 2015. https://doi.org/10.1016/j.amc.2015.02.085
S.A. Messaoudi and N. e. Tatar. Uniform stabilization of solutions of a nonlinear system of viscoelastic equations. Appl. Anal., 87(3):247–263, 2008. https://doi.org/10.1080/00036810701668394
S.A. Messaoudi and J.H. Hassan. On the general decay for a system of viscoelastic wave equations. In: Dutta H., Koinac L., Srivastava H. (eds) Current Trends in Mathematical Analysis and Its Interdisciplinary Applications, 2019. https://doi.org/10.1007/978-1-4757-2063-1
M.I. Mustafa. Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations. Nonlinear Anal. Real World Appl., 13(1):452– 463, 2012. https://doi.org/10.1016/j.nonrwa.2011.08.002
M.I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete Contin. Dyn. Syst. Ser. A., 35(3):1179–1192, 2015. https://doi.org/10.3934/dcds.2015.35.1179
M.I. Mustafa. Uniform decay rates for viscoelastic dissipative systems. J. Dyn. Control Syst., 22(1):101–116, 2016. https://doi.org/10.1007/s10883-014-9256-1
M.I. Mustafa. General decay result for nonlinear viscoelastic equations. J. Math. Anal. Appl., 457(1):134–152, 2018. https://doi.org/10.1016/j.jmaa.2017.08.019
M.I. Mustafa. Optimal decay rates for the viscoelastic wave equation. Math. Methods Appl. Sci., 41:192–204, 2018. https://doi.org/10.1002/mma.4604
M.I. Mustafa and S.A. Messaoudi. General stability result for viscoelastic wave equations. J. Math. Phys., 53(5), 2012. https://doi.org/10.1063/1.4711830
J. Park and S. Park. General decay for quasilinear viscoelastic equations with nonlinear weak damping. J. Math. Phys., 50(8), 2009. https://doi.org/10.1063/1.3187780
B. Said-Houari, S.A. Messaoudi and A. Guesmia. General decay of solutions of a nonlinear system of viscoelastic wave equations. NoDEA Nonlinear Differ. Equ. Appl., 18:659–684, 2011. https://doi.org/10.1007/s00030-011-0112-7
T.J. Xiao and J.Liang. Coupled second order semilinear evolution equations indirectly damped via memory effects. J. Differ. Equ., 254(5):2128–2157, 2013. https://doi.org/10.1016/j.jde.2012.11.019
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(15–16):867–872, 2009. https://doi.org/10.1016/j.crma.2009.05.011
V.I. Arnold. Mathematical Methods of Classical Mechanics. Springer-Verlag, New York, 1989. https://doi.org/10.1007/978-1-4757-2063-1
A. Bchatnia and M. Daoulatli. Behavior of the energy for Lam´e systems in bounded domains with nonlinear damping and external force. Electron. J. Differential Equations, 2013:1–17, 2013.
A. Bchatnia and A. Guesmia. Well-posedness and asymptotic stability for the Lam´e system with infinite memories in a bounded domain. Math. Control Relat. Fields, 4(4):451–463, 2014. https://doi.org/10.3934/mcrf.2014.4.451
A. Beniani, N. Taouaf and A. Benaissa. Well-posedness and exponential stability for coupled Lam´e system with viscoelastic term and strong damping. Computers Math. Appl., 75(12):4397–4404, 2018. https://doi.org/10.1016/j.camwa.2018.03.037
A. Beniani, K. Zennir and A. Benaissa. Stability for the Lam´e system with a time varying delay term in a nonlinear internal feedback. Clifford Anal. Clifford Algebr. Appl., 5:287–298, 2016.
M.M. Cavalcanti, V.N.D. Cavalcanti, I. Lasiecka and F.A. Nascimento. Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects. Discrete Conti. Dyn. Syst. Ser. B, 19:1987–2012, 2014. https://doi.org/10.3934/dcdsb.2014.19.1987
M.M. Cavalcanti, V.N.D. Cavalcanti, I. Lasiecka and X. Wang. Existence and sharp decay rate estimates for a von Karman system with long memory. Nonlinear Anal.: Real World Appl., 22:289–306, 2015. https://doi.org/10.1016/j.nonrwa.2014.09.016
X. Han and M. Wang. General decay of energy for a viscoelastic equation with nonlinear damping. Math. Methods Appl. Sci., 32:346–358, 2009. https://doi.org/10.1002/mma.1041
I. Lasiecka, S.A. Messaoudi and M.I. Mustafa. Note on intrinsic decay rates for abstract wave equations with memory. J. Math. Phys., 54(3), 2013. https://doi.org/10.1063/1.4793988
I. Lasiecka and D. Tataru. Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping. Differential Integral Equations, 8:507–533, 1993.
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, Springer INdAM Series, 10:271–303, 2014.
W.J. Liu. General decay of solutions to a viscoelastic wave equation with nonlinear localized damping. Ann. Acad. Sci. Fenn. Math., 34:291–302, 2009.
W.J. Liu. General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms. J. Math. Phys., 50:113506, 2009. https://doi.org/10.1063/1.3254323
S.A. Messaoud and M.I. Mustafa. On the control of solutions of viscoelastic equations with boundary feedback. Nonlinear Anal.: Real World Appl., 10(5):3132– 3140, 2009. https://doi.org/10.1016/j.nonrwa.2008.10.026
S.A. Messaoudi. General decay of solutions of a viscoelastic equation. J. Math. Anal. Appl., 341(2):1457–1467, 2008. https://doi.org/10.1016/j.jmaa.2007.11.048
S.A. Messaoudi. General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Anal., 69(8):2589–2598, 2008. https://doi.org/10.1016/j.na.2007.08.035
S.A. Messaoudi and M.M. Al-Gharabli. A general decay result of a nonlinear system of wave equations with infinite memories. Appl. Math. Compu., 259(15):540– 551, 2015. https://doi.org/10.1016/j.amc.2015.02.085
S.A. Messaoudi and N. e. Tatar. Uniform stabilization of solutions of a nonlinear system of viscoelastic equations. Appl. Anal., 87(3):247–263, 2008. https://doi.org/10.1080/00036810701668394
S.A. Messaoudi and J.H. Hassan. On the general decay for a system of viscoelastic wave equations. In: Dutta H., Koinac L., Srivastava H. (eds) Current Trends in Mathematical Analysis and Its Interdisciplinary Applications, 2019. https://doi.org/10.1007/978-1-4757-2063-1
M.I. Mustafa. Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations. Nonlinear Anal. Real World Appl., 13(1):452– 463, 2012. https://doi.org/10.1016/j.nonrwa.2011.08.002
M.I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete Contin. Dyn. Syst. Ser. A., 35(3):1179–1192, 2015. https://doi.org/10.3934/dcds.2015.35.1179
M.I. Mustafa. Uniform decay rates for viscoelastic dissipative systems. J. Dyn. Control Syst., 22(1):101–116, 2016. https://doi.org/10.1007/s10883-014-9256-1
M.I. Mustafa. General decay result for nonlinear viscoelastic equations. J. Math. Anal. Appl., 457(1):134–152, 2018. https://doi.org/10.1016/j.jmaa.2017.08.019
M.I. Mustafa. Optimal decay rates for the viscoelastic wave equation. Math. Methods Appl. Sci., 41:192–204, 2018. https://doi.org/10.1002/mma.4604
M.I. Mustafa and S.A. Messaoudi. General stability result for viscoelastic wave equations. J. Math. Phys., 53(5), 2012. https://doi.org/10.1063/1.4711830
J. Park and S. Park. General decay for quasilinear viscoelastic equations with nonlinear weak damping. J. Math. Phys., 50(8), 2009. https://doi.org/10.1063/1.3187780
B. Said-Houari, S.A. Messaoudi and A. Guesmia. General decay of solutions of a nonlinear system of viscoelastic wave equations. NoDEA Nonlinear Differ. Equ. Appl., 18:659–684, 2011. https://doi.org/10.1007/s00030-011-0112-7
T.J. Xiao and J.Liang. Coupled second order semilinear evolution equations indirectly damped via memory effects. J. Differ. Equ., 254(5):2128–2157, 2013. https://doi.org/10.1016/j.jde.2012.11.019