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On global axisymmetric solutions to 2D compressible full Euler equations of Chaplygin gases
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 |
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.
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. |
[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-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. |
[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. |
[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.
|
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
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. |
[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-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. |
[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. |
[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.
|
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
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