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Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions
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 |
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.
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. |
[60] |
F. G. Shinskey, Process Control Systems, McGraw-Hill Book Company, New York, 1967. |
[61] |
J. Simon,
Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[62] |
I. H. Suh and Z. Bien,
Use of time delay action in the controller design, IEEE Transactions on Automatic Control, 25 (1980), 600-603.
|
[63] |
S. T. Wu,
Asymptotic Behavior for a Viscoelastic Wave Equation with a Delay Term, Taiwanese Journal of Mathematics, 17 (1980), 765-784.
doi: 10.11650/tjm.17.2013.2517. |
[64] |
C. Q. Xu, S. P. Yung and L. K. Li,
Stabilization of the wave system with input delay in the boundary control, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 770-785.
doi: 10.1051/cocv:2006021. |
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. |
[60] |
F. G. Shinskey, Process Control Systems, McGraw-Hill Book Company, New York, 1967. |
[61] |
J. Simon,
Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[62] |
I. H. Suh and Z. Bien,
Use of time delay action in the controller design, IEEE Transactions on Automatic Control, 25 (1980), 600-603.
|
[63] |
S. T. Wu,
Asymptotic Behavior for a Viscoelastic Wave Equation with a Delay Term, Taiwanese Journal of Mathematics, 17 (1980), 765-784.
doi: 10.11650/tjm.17.2013.2517. |
[64] |
C. Q. Xu, S. P. Yung and L. K. Li,
Stabilization of the wave system with input delay in the boundary control, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 770-785.
doi: 10.1051/cocv:2006021. |
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