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June  2021, 11(2): 353-371. doi: 10.3934/mcrf.2020040

General stability of abstract thermoelastic system with infinite memory and delay

Jianghao Hao , and  Junna Zhang

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi, 030006, China

Corresponding author: Jianghao Hao

Received  January 2020 Revised  July 2020 Published  June 2021 Early access  October 2020

Fund Project: This research was partially supported by Natural Science Foundation of China (grant number 11871315, 61374089), Natural Science Foundation of Shanxi Province of China (grant number 201801D121003, 201901D111021)

In this paper we study an abstract thermoelastic system in Hilbert space with infinite memory and time delay. Under some suitable conditions, we prove the well-posedness by invoking semigroup theory. Since the damping may stabilize the system while the delay may destabilize it, we discuss the interaction between the damping and the delay term, and obtain that the system is uniformly stable when the effect of damping is greater than that of time delay. By establishing suitable Lyapunov functionals which are equivalent to the energy of system we also establish the general energy decay results for abstract thermoelastic system.

Keywords: Abstract thermoelastic system, infinite memory, time delay, well-posedness, general stability.
Mathematics Subject Classification: Primary: 35B35, 93D15; Secondary: 80A17.
Citation: Jianghao Hao, Junna Zhang. General stability of abstract thermoelastic system with infinite memory and delay. Mathematical Control and Related Fields, 2021, 11 (2) : 353-371. doi: 10.3934/mcrf.2020040
References:
[1]

M. Afilal and A. Soufyane, General decay for a porous thermoelastic system with a memory, Appl. Anal., 98 (2019), 638-650.  doi: 10.1080/00036811.2017.1399363.

[2]

F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Math. Acad. Sci. Paris, 347 (2009), 867-872.  doi: 10.1016/j.crma.2009.05.011.

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F. Alabau-Boussouira, S. Nicaise and C. Pignotti, Exponential stability of the wave equation with memory and time delay, in New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer INdAM Ser., 10, Springer, Cham, 2014, 1–22. doi: 10.1007/978-3-319-11406-4_1.

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H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.

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C. Buriol, Energy decay rates for the Timoshenko system of thermoelastic plates, Nonlinear Anal., 64 (2006), 92-108.  doi: 10.1016/j.na.2005.06.010.

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

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E. Fridman, S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with time-dependent delay, SIAM J. Control Optim., 48 (2010), 5028-5052.  doi: 10.1137/090762105.

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A. Guesmia, Asymptotic stability of abstract dissipative systems with infinite memory, J. Math. Anal. Appl., 382 (2011), 748-760.  doi: 10.1016/j.jmaa.2011.04.079.

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A. Guesmia, Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay, IMA J. Math. Control Inform., 30 (2013), 507-526.  doi: 10.1093/imamci/dns039.

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A. Guesmia and S. A. Messaoudi, A new approach to the stability of an abstract system in the presence of infinite history, J. Math. Anal. Appl., 416 (2014), 212-228.  doi: 10.1016/j.jmaa.2014.02.030.

[11]

J. Hao and P. Wang, General stability result of abstract thermoelastic system with infinite memory, Bull. Malays. Math. Sci. Soc., 42 (2019), 2549-2567.  doi: 10.1007/s40840-018-0615-z.

[12]

K.-P. Jin, J. Liang and T.-J. Xiao, Asymptotic behavior for coupled systems of second order abstract evolution equations with one infinite memory, J. Math. Anal. Appl., 475 (2019), 554-575.  doi: 10.1016/j.jmaa.2019.02.055.

[13]

K.-P. Jin, J. Liang and T.-J. Xiao, Coupled second order evolution equations with fading memory: Optimal energy decay rate, J. Differential Equations, 257 (2014), 1501-1528.  doi: 10.1016/j.jde.2014.05.018.

[14]

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

[15]

I. Lasiecka, S. A. Messaoudi and M. I. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory, J. Math. Phys., 54 (2013), 18pp. doi: 10.1063/1.4793988.

[16]

M. J. Lee, J. Y. Park and Y. H. Kang, Asymptotic stability of a problem with Balakrishnan-Taylor damping and a time delay, Comput. Math. Appl., 70 (2015), 478-487.  doi: 10.1016/j.camwa.2015.05.004.

[17]

S. Mesloub and F. Mesloub, On a coupled nonlinear singular thermoelastic system, Nonlinear Anal., 73 (2010), 3195-3208.  doi: 10.1016/j.na.2010.06.082.

[18]

S. A. Messaoudi and W. Al-Khulaifi, General and optimal decay for a quasilinear viscoelastic equation, Appl. Math. Lett., 66 (2017), 16-22.  doi: 10.1016/j.aml.2016.11.002.

[19]

J. E. Muñoz Rivera and M. G. Naso, Asymptotic stability of semigroups associated with linear weak dissipative systems with memory, J. Math. Anal. Appl., 326 (2007), 691-707.  doi: 10.1016/j.jmaa.2006.03.022.

[20]

J. E. Muñoz Rivera, M. G. Naso and F. M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl., 286 (2003), 692-704.  doi: 10.1016/S0022-247X(03)00511-0.

[21]

J. E. Muñoz Rivera and M. G. Naso, Optimal energy decay rate for a class of weakly dissipative second-order systems with memory, Appl. Math. Lett., 23 (2010), 743-746.  doi: 10.1016/j.aml.2010.02.016.

[22]

M. I. Mustafa, Asymptotic behavior of second sound thermoelasticity with internal time-varying delay, Z. Angew. Math. Phys., 64 (2013), 1353-1362.  doi: 10.1007/s00033-012-0268-y.

[23]

S. Nicaise and C. Pignotti, Asymptotic stability of second-order evolution equations with intermittent delay, Adv. Differential Equations, 17 (2012), 879-902. 

[24]

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.

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[26]

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.

[27]

A. G. Ramm, Stability of solutions to abstract evolution equations with delay, J. Math. Anal. Appl., 396 (2012), 523-527.  doi: 10.1016/j.jmaa.2012.06.033.

[28]

Z. Yang, Existence and energy decay of solutions for the Euler-Bernoulli viscoelastic equation with a delay, Z. Angew. Math. Phys., 66 (2015), 727-745.  doi: 10.1007/s00033-014-0429-2.

[29]

A. Youkana, Stability of an abstract system with infinite history, preprint, arXiv: 1805.07964.

show all references

References:
[1]

M. Afilal and A. Soufyane, General decay for a porous thermoelastic system with a memory, Appl. Anal., 98 (2019), 638-650.  doi: 10.1080/00036811.2017.1399363.

[2]

F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Math. Acad. Sci. Paris, 347 (2009), 867-872.  doi: 10.1016/j.crma.2009.05.011.

[3]

F. Alabau-Boussouira, S. Nicaise and C. Pignotti, Exponential stability of the wave equation with memory and time delay, in New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer INdAM Ser., 10, Springer, Cham, 2014, 1–22. doi: 10.1007/978-3-319-11406-4_1.

[4]

H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.

[5]

C. Buriol, Energy decay rates for the Timoshenko system of thermoelastic plates, Nonlinear Anal., 64 (2006), 92-108.  doi: 10.1016/j.na.2005.06.010.

[6]

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

[7]

E. Fridman, S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with time-dependent delay, SIAM J. Control Optim., 48 (2010), 5028-5052.  doi: 10.1137/090762105.

[8]

A. Guesmia, Asymptotic stability of abstract dissipative systems with infinite memory, J. Math. Anal. Appl., 382 (2011), 748-760.  doi: 10.1016/j.jmaa.2011.04.079.

[9]

A. Guesmia, Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay, IMA J. Math. Control Inform., 30 (2013), 507-526.  doi: 10.1093/imamci/dns039.

[10]

A. Guesmia and S. A. Messaoudi, A new approach to the stability of an abstract system in the presence of infinite history, J. Math. Anal. Appl., 416 (2014), 212-228.  doi: 10.1016/j.jmaa.2014.02.030.

[11]

J. Hao and P. Wang, General stability result of abstract thermoelastic system with infinite memory, Bull. Malays. Math. Sci. Soc., 42 (2019), 2549-2567.  doi: 10.1007/s40840-018-0615-z.

[12]

K.-P. Jin, J. Liang and T.-J. Xiao, Asymptotic behavior for coupled systems of second order abstract evolution equations with one infinite memory, J. Math. Anal. Appl., 475 (2019), 554-575.  doi: 10.1016/j.jmaa.2019.02.055.

[13]

K.-P. Jin, J. Liang and T.-J. Xiao, Coupled second order evolution equations with fading memory: Optimal energy decay rate, J. Differential Equations, 257 (2014), 1501-1528.  doi: 10.1016/j.jde.2014.05.018.

[14]

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

[15]

I. Lasiecka, S. A. Messaoudi and M. I. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory, J. Math. Phys., 54 (2013), 18pp. doi: 10.1063/1.4793988.

[16]

M. J. Lee, J. Y. Park and Y. H. Kang, Asymptotic stability of a problem with Balakrishnan-Taylor damping and a time delay, Comput. Math. Appl., 70 (2015), 478-487.  doi: 10.1016/j.camwa.2015.05.004.

[17]

S. Mesloub and F. Mesloub, On a coupled nonlinear singular thermoelastic system, Nonlinear Anal., 73 (2010), 3195-3208.  doi: 10.1016/j.na.2010.06.082.

[18]

S. A. Messaoudi and W. Al-Khulaifi, General and optimal decay for a quasilinear viscoelastic equation, Appl. Math. Lett., 66 (2017), 16-22.  doi: 10.1016/j.aml.2016.11.002.

[19]

J. E. Muñoz Rivera and M. G. Naso, Asymptotic stability of semigroups associated with linear weak dissipative systems with memory, J. Math. Anal. Appl., 326 (2007), 691-707.  doi: 10.1016/j.jmaa.2006.03.022.

[20]

J. E. Muñoz Rivera, M. G. Naso and F. M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl., 286 (2003), 692-704.  doi: 10.1016/S0022-247X(03)00511-0.

[21]

J. E. Muñoz Rivera and M. G. Naso, Optimal energy decay rate for a class of weakly dissipative second-order systems with memory, Appl. Math. Lett., 23 (2010), 743-746.  doi: 10.1016/j.aml.2010.02.016.

[22]

M. I. Mustafa, Asymptotic behavior of second sound thermoelasticity with internal time-varying delay, Z. Angew. Math. Phys., 64 (2013), 1353-1362.  doi: 10.1007/s00033-012-0268-y.

[23]

S. Nicaise and C. Pignotti, Asymptotic stability of second-order evolution equations with intermittent delay, Adv. Differential Equations, 17 (2012), 879-902. 

[24]

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.

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[26]

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.

[27]

A. G. Ramm, Stability of solutions to abstract evolution equations with delay, J. Math. Anal. Appl., 396 (2012), 523-527.  doi: 10.1016/j.jmaa.2012.06.033.

[28]

Z. Yang, Existence and energy decay of solutions for the Euler-Bernoulli viscoelastic equation with a delay, Z. Angew. Math. Phys., 66 (2015), 727-745.  doi: 10.1007/s00033-014-0429-2.

[29]

A. Youkana, Stability of an abstract system with infinite history, preprint, arXiv: 1805.07964.

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