March  2020, 40(3): 1493-1515. doi: 10.3934/dcds.2020084

Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay

1. 

College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China

2. 

Department of Mathematics and Economics, Virginia State University, Petersburg, VA 23806, USA

3. 

College of Applied Science, Beijing University of Technology, Beijing 100124, China

* Corresponding author: Jing Zhang

Received  February 2019 Revised  October 2019 Published  December 2019

Fund Project: Research partly supported by NSFC (No. 11831003, No. 11771031, No. 11531010 and No. 11726625), the Fund of Young Backbone Teacher in Henan Province (No. 2018GGJS039).

This paper is concerned with the stability and dynamics of a weak viscoelastic system with nonlinear time-varying delay. By imposing appropriate assumptions on the memory and sub-linear delay operator, we prove the global well-posedness and stability which generates a gradient system. The gradient system possesses finite fractal dimensional global and exponential attractors with unstable manifold structure. Moreover, the effect and balance between damping and time-varying delay are also presented.

Citation: Xin-Guang Yang, Jing Zhang, Shu Wang. Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1493-1515. doi: 10.3934/dcds.2020084
References:
[1]

M. AassilaM. M. Cavalcanti and J. A. Soriano, Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain, SIAM J. Control Optim., 38 (2000), 1581-1602.  doi: 10.1137/S0363012998344981.

[2]

C. Abdallah, P. Dorato and R. Byrne, Delayed positive feedback can stabilize oscillatory system, American Control Conference, San Francisco, 1993, 3106–3107. doi: 10.23919/ACC.1993.4793475.

[3]

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.

[4]

J. ApplebyM. FabrizioB. Lazzari and D. Reynolds, On exponential asymptotic stability in linear viscoelasticity, Math. Models Methods Appl. Sci., 16 (2006), 1677-1694.  doi: 10.1142/S0218202506001674.

[5]

M. M. CavalcantiD. V. N. Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differential Equations, 2002 (2002), 1-14. 

[6]

I. D. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., (195) (2008). doi: 10.1090/memo/0912.

[7]

I. D. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-Posedness and Long Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[8]

M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720.  doi: 10.3934/cpaa.2005.4.705.

[9]

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

[10]

Q. Dai and Z. Yang, Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 65 (2014), 885-903.  doi: 10.1007/s00033-013-0365-6.

[11]

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

[12]

M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal., 81 (2002), 1245-1264.  doi: 10.1080/0003681021000035588.

[13]

B. Feng, General decay for a viscoelastic wave equation with density and time delay term in $\mathbb{R}^n$, Taiwanese J. Math., 22 (2018), 205-223.  doi: 10.11650/tjm/8105.

[14]

E. Fridman, Introduction to Time-Delay Systems. Analysis and Control, Systems & Control: Foundations & Applications, Birkhäser/Springer, Cham, 2014. doi: 10.1007/978-3-319-09393-2.

[15]

C. GiorgiJ. Muñoz Rivera and V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83-99.  doi: 10.1006/jmaa.2001.7437.

[16]

A. Guesmia and S. A. Messaoudi, A general decay result for a viscoelastic equation in the presence of past and finite history memories, Nonlinear Anal. Real World Appl., 13 (2012), 476-485.  doi: 10.1016/j.nonrwa.2011.08.004.

[17]

Y. Guo, M. A. Rammaha and S. Sakuntasathien, Energy decay of a viscoelastic wave equation with supercritical nonlinearities, Z. Angew. Math. Phys., 69 (2018), 28pp. doi: 10.1007/s00033-018-0961-6.

[18]

M. Kirane and B. Said-Houari, Existence and asymptotic stability of viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082.  doi: 10.1007/s00033-011-0145-0.

[19]

W. Liu, General decay rate estimate for the energy of a weak viscoealstic equation with internal time-varying delay term, Taiwanese J. Math., 17 (2013), 2101-2115.  doi: 10.11650/tjm.17.2013.2968.

[20]

G. Liu and L. Diao, Energy decay of the solution for a weak viscoelastic equation with a time-varying delay, Acta. Appl. Math., 155 (2018), 9-19.  doi: 10.1007/s10440-017-0142-1.

[21]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Research Notes in Mathematics, 398, Chapman Hall & CRC, Boca Raton, FL, 1999.

[22]

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

[23]

S. A. Messaoudi, General decay of solutions of a weak viscoelastic equation, Arab. J. Sci. Eng., 36 (2011), 1569-1579.  doi: 10.1007/s13369-011-0132-y.

[24]

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

[25]

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

[26]

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

[27]

S. NicaiseC. Pignotti and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559.

[28]

S. NicaiseC. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693.

[29]

V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360.  doi: 10.1007/s00032-009-0098-3.

[30]

C. Pignotti, Stability results for second-order evolution equations with memory and switching time-delay, J. Dynam. Differential Equations, 29 (2017), 1309-1324.  doi: 10.1007/s10884-016-9545-3.

[31]

Y. QinJ. Ren and T. Wei, Global existence, asymptotic behavior, and uniform attractor for a nonautonomous equation, Math. Methods Appl. Sci., 36 (2013), 2540-2553.  doi: 10.1002/mma.2774.

[32]

B. Said-Houari, Asymptotic behaviors of solutions for viscoelastic wave equation with space-time dependent damping term, J. Math. Anal. Appl., 387 (2012), 1088-1105.  doi: 10.1016/j.jmaa.2011.10.017.

[33]

I. H. Suh and Z. Bien, Use of time delay actions in the controller design, IEEE Trans. Automatic Control, 25 (1980), 600-603.  doi: 10.1109/TAC.1980.1102347.

[34]

C. Q. XuS. P. Yung and L. K. Li, Stabilization of the wave system with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.  doi: 10.1051/cocv:2006021.

show all references

References:
[1]

M. AassilaM. M. Cavalcanti and J. A. Soriano, Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain, SIAM J. Control Optim., 38 (2000), 1581-1602.  doi: 10.1137/S0363012998344981.

[2]

C. Abdallah, P. Dorato and R. Byrne, Delayed positive feedback can stabilize oscillatory system, American Control Conference, San Francisco, 1993, 3106–3107. doi: 10.23919/ACC.1993.4793475.

[3]

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.

[4]

J. ApplebyM. FabrizioB. Lazzari and D. Reynolds, On exponential asymptotic stability in linear viscoelasticity, Math. Models Methods Appl. Sci., 16 (2006), 1677-1694.  doi: 10.1142/S0218202506001674.

[5]

M. M. CavalcantiD. V. N. Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differential Equations, 2002 (2002), 1-14. 

[6]

I. D. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., (195) (2008). doi: 10.1090/memo/0912.

[7]

I. D. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-Posedness and Long Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[8]

M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720.  doi: 10.3934/cpaa.2005.4.705.

[9]

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

[10]

Q. Dai and Z. Yang, Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 65 (2014), 885-903.  doi: 10.1007/s00033-013-0365-6.

[11]

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

[12]

M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal., 81 (2002), 1245-1264.  doi: 10.1080/0003681021000035588.

[13]

B. Feng, General decay for a viscoelastic wave equation with density and time delay term in $\mathbb{R}^n$, Taiwanese J. Math., 22 (2018), 205-223.  doi: 10.11650/tjm/8105.

[14]

E. Fridman, Introduction to Time-Delay Systems. Analysis and Control, Systems & Control: Foundations & Applications, Birkhäser/Springer, Cham, 2014. doi: 10.1007/978-3-319-09393-2.

[15]

C. GiorgiJ. Muñoz Rivera and V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83-99.  doi: 10.1006/jmaa.2001.7437.

[16]

A. Guesmia and S. A. Messaoudi, A general decay result for a viscoelastic equation in the presence of past and finite history memories, Nonlinear Anal. Real World Appl., 13 (2012), 476-485.  doi: 10.1016/j.nonrwa.2011.08.004.

[17]

Y. Guo, M. A. Rammaha and S. Sakuntasathien, Energy decay of a viscoelastic wave equation with supercritical nonlinearities, Z. Angew. Math. Phys., 69 (2018), 28pp. doi: 10.1007/s00033-018-0961-6.

[18]

M. Kirane and B. Said-Houari, Existence and asymptotic stability of viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082.  doi: 10.1007/s00033-011-0145-0.

[19]

W. Liu, General decay rate estimate for the energy of a weak viscoealstic equation with internal time-varying delay term, Taiwanese J. Math., 17 (2013), 2101-2115.  doi: 10.11650/tjm.17.2013.2968.

[20]

G. Liu and L. Diao, Energy decay of the solution for a weak viscoelastic equation with a time-varying delay, Acta. Appl. Math., 155 (2018), 9-19.  doi: 10.1007/s10440-017-0142-1.

[21]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Research Notes in Mathematics, 398, Chapman Hall & CRC, Boca Raton, FL, 1999.

[22]

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

[23]

S. A. Messaoudi, General decay of solutions of a weak viscoelastic equation, Arab. J. Sci. Eng., 36 (2011), 1569-1579.  doi: 10.1007/s13369-011-0132-y.

[24]

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

[25]

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

[26]

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

[27]

S. NicaiseC. Pignotti and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559.

[28]

S. NicaiseC. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693.

[29]

V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360.  doi: 10.1007/s00032-009-0098-3.

[30]

C. Pignotti, Stability results for second-order evolution equations with memory and switching time-delay, J. Dynam. Differential Equations, 29 (2017), 1309-1324.  doi: 10.1007/s10884-016-9545-3.

[31]

Y. QinJ. Ren and T. Wei, Global existence, asymptotic behavior, and uniform attractor for a nonautonomous equation, Math. Methods Appl. Sci., 36 (2013), 2540-2553.  doi: 10.1002/mma.2774.

[32]

B. Said-Houari, Asymptotic behaviors of solutions for viscoelastic wave equation with space-time dependent damping term, J. Math. Anal. Appl., 387 (2012), 1088-1105.  doi: 10.1016/j.jmaa.2011.10.017.

[33]

I. H. Suh and Z. Bien, Use of time delay actions in the controller design, IEEE Trans. Automatic Control, 25 (1980), 600-603.  doi: 10.1109/TAC.1980.1102347.

[34]

C. Q. XuS. P. Yung and L. K. Li, Stabilization of the wave system with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.  doi: 10.1051/cocv:2006021.

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