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

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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

Ammar Khemmoudj ^, and Yacine Mokhtari

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.

Keywords: Nonlinear von Kármán system, viscoelastic term, delay term, decay rate, Lyapunov functionals.

Mathematics Subject Classification: Primary: 35B40, 35L70, 49K25, 93D15, 93D20, 74K10.

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

References:

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[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.
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[9]	A. Benaissa, A. 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.
[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.
[11]	H. Brezis, Analyse Fonctionnelle: Theéorie et Applications, Dunod, Paris, 1983.
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[21]	R. Datko, J. 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.
[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.
[23]	J. F. Doyle, Wave Propagation in Structures, Springer-Verlag, New York, 1997.
[24]	M. Eller, J. 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.
[25]	A. Favini, M. A. Horn, I. 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.
[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.
[27]	G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities,, Cambridge University Press, Cambridge, 1988.
[28]	M. Kafini, S. 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.
[29]	V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, Masson-John Wiley, Paris, 1994.
[30]	J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM, 1989. doi: 10.1137/1.9781611970821.
[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.
[32]	J. E. Lagnese and J. L. Lions, Modelling Analysis and Control of Thin Plates, RMA 6, Masson, Paris, 1988.
[33]	J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia, PA., 1989. doi: 10.1137/1.9781611970821.
[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.
[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.
[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.
[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.
[38]	M. J. Lee, J. 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.
[39]	J. L. Lions, Quelques Methodes de Resolution des Problemes Aux Limites non Lineaires, Dunod, Paris, (in French) 1969.
[40]	W. J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75.
[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.
[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.
[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.
[44]	J. E. Munoz Rivera, H. 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[52]	S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differ. Integral Equ., 21 (2008), 935-958.
[53]	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.
[54]	S. H. Park, J. 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.
[55]	J. Y. Park and S. H. Park, General decay for quasilinear viscoelastic equations with nonlinear weak damping, Journal of Mathematical Physics, 50 (2009), Article ID: 083505, 10pp. doi: 10.1063/1.3187780.
[56]	G. Perla Menzala and E. Zuazua, On a one-dimensional version of the dynamical Marguerre-Vlasov system, Bol. Soc. Bras. Mat., 32 (2001), 303-319. doi: 10.1007/BF01233669.
[57]	G. Perla Menzala and E. Zuazua, The beam equation as a limit of a 1-D nonlinear von Kármán model, Applied Mathematics Letters, 12 (1999), 47-52. doi: 10.1016/S0893-9659(98)00125-6.
[58]	G. Perla Menzala and E. Zuazua, Timoshenko's plate equation as a singular limit of the dynamical von Kármán system, J. Math. Pures Appl., 79 (2000), 73-94. doi: 10.1016/S0021-7824(00)00149-5.
[59]	G. Perla Menzala and E. Zuazua, Timoshenko's beam equation as a limit of a nonlinear one-dimentional von Kármán system, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 855-875. doi: 10.1017/S0308210500000470.
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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.
[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.
[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.
[4]	F. D. Araruna, P. 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.
[5]	V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.
[6]	V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.
[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.
[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.
[9]	A. Benaissa, A. 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.
[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.
[11]	H. Brezis, Analyse Fonctionnelle: Theéorie et Applications, Dunod, Paris, 1983.
[12]	M. M. Cavalcanti, V. 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.
[13]	M. M. Cavalcanti, V. 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.
[14]	M. M. Cavalcanti, V. 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.
[15]	M. M. Cavalcanti, V. 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.
[16]	I. Chueshov, M. 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.
[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.
[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.
[19]	C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. doi: 10.1007/BF00251609.
[20]	M. Daoulatli, I. 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.
[21]	R. Datko, J. 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.
[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.
[23]	J. F. Doyle, Wave Propagation in Structures, Springer-Verlag, New York, 1997.
[24]	M. Eller, J. 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.
[25]	A. Favini, M. A. Horn, I. 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.
[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.
[27]	G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities,, Cambridge University Press, Cambridge, 1988.
[28]	M. Kafini, S. 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.
[29]	V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, Masson-John Wiley, Paris, 1994.
[30]	J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM, 1989. doi: 10.1137/1.9781611970821.
[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.
[32]	J. E. Lagnese and J. L. Lions, Modelling Analysis and Control of Thin Plates, RMA 6, Masson, Paris, 1988.
[33]	J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia, PA., 1989. doi: 10.1137/1.9781611970821.
[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.
[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.
[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.
[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.
[38]	M. J. Lee, J. 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.
[39]	J. L. Lions, Quelques Methodes de Resolution des Problemes Aux Limites non Lineaires, Dunod, Paris, (in French) 1969.
[40]	W. J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75.
[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.
[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.
[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.
[44]	J. E. Munoz Rivera, H. 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[52]	S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differ. Integral Equ., 21 (2008), 935-958.
[53]	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.
[54]	S. H. Park, J. 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.
[55]	J. Y. Park and S. H. Park, General decay for quasilinear viscoelastic equations with nonlinear weak damping, Journal of Mathematical Physics, 50 (2009), Article ID: 083505, 10pp. doi: 10.1063/1.3187780.
[56]	G. Perla Menzala and E. Zuazua, On a one-dimensional version of the dynamical Marguerre-Vlasov system, Bol. Soc. Bras. Mat., 32 (2001), 303-319. doi: 10.1007/BF01233669.
[57]	G. Perla Menzala and E. Zuazua, The beam equation as a limit of a 1-D nonlinear von Kármán model, Applied Mathematics Letters, 12 (1999), 47-52. doi: 10.1016/S0893-9659(98)00125-6.
[58]	G. Perla Menzala and E. Zuazua, Timoshenko's plate equation as a singular limit of the dynamical von Kármán system, J. Math. Pures Appl., 79 (2000), 73-94. doi: 10.1016/S0021-7824(00)00149-5.
[59]	G. Perla Menzala and E. Zuazua, Timoshenko's beam equation as a limit of a nonlinear one-dimentional von Kármán system, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 855-875. doi: 10.1017/S0308210500000470.
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