American Institute of Mathematical Sciences

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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

Xin-Guang Yang 1, , Jing Zhang 2,, and  Shu Wang 3,

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.

Keywords: Viscoelastic system, time varying delay, quasi-stability, unstable manifold, exponential attractor.
Mathematics Subject Classification: 35B40, 35B41, 35Q30, 76D03, 76D05.
Citation: Xin-Guang Yang, Jing Zhang, Shu Wang. Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1493-1515. doi: 10.3934/dcds.2020084
References:
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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.  Google Scholar

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S. Nicaise, C. 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.  Google Scholar

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S. Nicaise, C. 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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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C. Q. Xu, S. 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.  Google Scholar

show all references

References:
[1]

M. Aassila, M. 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.  Google Scholar

[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.  Google Scholar

[3]

F. Alabau-Boussouira, P. 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.  Google Scholar

[4]

J. Appleby, M. Fabrizio, B. Lazzari and D. Reynolds, On exponential asymptotic stability in linear viscoelasticity, Math. Models Methods Appl. Sci., 16 (2006), 1677-1694.  doi: 10.1142/S0218202506001674.  Google Scholar

[5]

M. M. Cavalcanti, D. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[11]

R. Datko, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

C. Giorgi, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

S. Nicaise, C. 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.  Google Scholar

[28]

S. Nicaise, C. 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.  Google Scholar

[29]

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

[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.  Google Scholar

[31]

Y. Qin, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

C. Q. Xu, S. 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.  Google Scholar

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