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doi: 10.3934/eect.2022027
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Theoretical and computational decay results for a Bresse system with one infinite memory in the longitudinal displacement

1. 

Arts et Metiers Institute of Technology, Centrale Lille, Junia, ULR2697 - L2EP, University of Lille, France

2. 

The Preparatory Year Program, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

3. 

IRC for construction and building materials, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

* Corresponding author: Adel M. Al-Mahdi

Received  July 2021 Revised  March 2022 Early access May 2022

Fund Project: The second and third authors are supported by KFUPM under Project No. SB201012

In this paper, we consider a one-dimensional linear Bresse system with only one infinite memory term acting in the third equation (longitudinal displacements). Under a general condition on the memory kernel (relaxation function), we establish a decay estimate of the energy of the system. Our decay result extends and improves some decay rates obtained in the literature such as the one in [27], [4], [33], [58] and [34]. The proof is based on the energy method together with convexity arguments. Numerical simulations are given to illustrate the theoretical decay result.

Citation: Mohamed Alahyane, Mohammad M. Al-Gharabli, Adel M. Al-Mahdi. Theoretical and computational decay results for a Bresse system with one infinite memory in the longitudinal displacement. Evolution Equations and Control Theory, doi: 10.3934/eect.2022027
References:
[1]

M. AfilalA. GuesmiaA. Soufyane and M. Zahri, On the exponential and polynomial stability for a linear Bresse system, Math. Methods Appl. Sci., 43 (2020), 2626-2645.  doi: 10.1002/mma.6070.

[2]

A. M. Al-Mahdi, General stability result for a viscoelastic plate equation with past history and general kernel, J. Math. Anal. Appl., 490 (2020), 124216, 19 pp. doi: 10.1016/j.jmaa.2020.124216.

[3]

A. M. Al-Mahdi, Stability result of a viscoelastic plate equation with past history and a logarithmic nonlinearity, Bound. Value Probl., 2020 (2020), Paper No. 84, 20 pp. doi: 10.1186/s13661-020-01382-9.

[4]

A. M. Al-Mahdi and M. M. Al-Gharabli, New general decay results in an infinite memory viscoelastic problem with nonlinear damping, Bound. Value Probl., 2019 (2019), Paper No. 140, 15 pp. doi: 10.1186/s13661-019-1253-6.

[5]

A. M. Al-MahdiM. M. Al-Gharabli and S. M. Ali, New stability result for a bresse system with one infinite memory in the shear angle equation, Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), 995-1014.  doi: 10.3934/dcdss.2021086.

[6]

A. M. Al-Mahdi, M. M. Al-Gharabli, A. Guesmia and S. A. Messaoudi, New decay results for a viscoelastic-type Timoshenko system with infinite memory, Z. Angew. Math. Phys., 72 (2021), Paper No. 22, 24 pp. doi: 10.1007/s00033-020-01446-x.

[7]

A. M. Al-Mahdi, M. M. Al-Gharabli, M. Kafini and S. Al-Omari, On the global existence and asymptotic behavior of the solution of a nonlinear wave equation with past history, J. Math. Phys., 62 (2021), Paper No. 031512, 15 pp. doi: 10.1063/5.0003515.

[8]

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.

[9]

F. Alabau BoussouiraJ. E. Muñoz Rivera and D. da S. Almeida Júnior, Stability to weak dissipative Bresse system, J. Math. Anal. Appl., 374 (2011), 481-498.  doi: 10.1016/j.jmaa.2010.07.046.

[10]

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

[11]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, vol. 60, Springer-Verlag, New York-Heidelberg, 1978.

[12]

L. Boltzmann, Zur theorie der elastischen nachwirkung, Annalen der Physik, 241 (1878), 430-432. 

[13]

J. A. C. Bresse, Cours de Mécanique Appliquée: Résistance des Matériaux et Stabilitédes Constructions, vol. 1, Mallet-Bachelier, 1859.

[14]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324.  doi: 10.1137/S0363012902408010.

[15]

B. Chentouf, A minimal state approach to dynamic stabilization of the rotating disk-beam system with infinite memory, IEEE Trans. Automat. Control, 61 (2016), 3700-3706.  doi: 10.1109/TAC.2016.2518482.

[16]

B. Chentouf and Z.-J. Han, On the elimination of infinite memory effects on the stability of a nonlinear non-homogeneous rotating body-beam system, Journal of Dynamics and Differential Equations, (2022), 1–25. doi: 10.1007/s10884-021-10111-4.

[17]

B. D. Coleman and V. J. Mizel, Norms and semi-groups in the theory of fading memory, Arch. Rational Mech. Anal., 23 (1966), 87-123.  doi: 10.1007/BF00251727.

[18]

B. D. Coleman and W. Noll, On the thermostatics of continuous media, Arch. Rational Mech. Anal., 4 (1959), 97-128.  doi: 10.1007/BF00281381.

[19]

B. D. Coleman and W. Noll, Foundations of linear viscoelasticity, Rev. Modern Phys., 33 (1961), 239-249.  doi: 10.1103/RevModPhys.33.239.

[20]

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.

[21]

C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations, 7 (1970), 554-569.  doi: 10.1016/0022-0396(70)90101-4.

[22]

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

[23]

G. Dassios and F. Zafiropoulos, Equipartition of energy in linearized $3$-D viscoelasticity, Quart. Appl. Math., 48 (1990), 715-730.  doi: 10.1090/qam/1079915.

[24]

L. H. Fatori and R. N. Monteiro, The optimal decay rate for a weak dissipative Bresse system, Appl. Math. Lett., 25 (2012), 600-604.  doi: 10.1016/j.aml.2011.09.067.

[25]

L. H. Fatori and J. E. M. Rivera, Rates of decay to weak thermoelastic Bresse system, IMA Journal of Applied Mathematics, 75 (2010), 881-904. 

[26]

C. GiorgiJ. E. 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.

[27]

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.

[28]

A. Guesmia, Asymptotic stability of Bresse system with one infinite memory in the longitudinal displacements, Mediterr. J. Math., 14 (2017), Paper No. 49, 19 pp. doi: 10.1007/s00009-017-0877-y.

[29]

A. Guesmia, Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement, Nonauton. Dyn. Syst., 4 (2017), 78-97.  doi: 10.1515/msds-2017-0008.

[30]

A. Guesmia, New general decay rates of solutions for two viscoelastic wave equations with infinite memory, Math. Model. Anal., 25 (2020), 351-373.  doi: 10.3846/mma.2020.10458.

[31]

A. Guesmia and M. Kafini, Bresse system with infinite memories, Math. Methods Appl. Sci., 38 (2015), 2389-2402.  doi: 10.1002/mma.3228.

[32]

A. Guesmia and M. Kirane, Uniform and weak stability of Bresse system with two infinite memories, Z. Angew. Math. Phys., 67 (2016), Art. 124, 39 pp. doi: 10.1007/s00033-016-0719-y.

[33]

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

[34]

A. Guesmia and N.-E. Tatar, Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay, Commun. Pure Appl. Anal., 14 (2015), 457-491.  doi: 10.3934/cpaa.2015.14.457.

[35]

X. Han and M. Wang, General decay estimate of energy for the second order evolution equations with memory, Acta Appl. Math., 110 (2010), 195-207.  doi: 10.1007/s10440-008-9397-x.

[36]

W. J. Hrusa, Global existence and asymptotic stability for a semilinear hyperbolic volterra equation with large initial data, SIAM J. Math. Anal., 16 (1985), 110-134.  doi: 10.1137/0516007.

[37]

J. E. LagneseG. Leugering and E. J. P. G. Schmidt, Modelling of dynamic networks of thin thermoelastic beams, Math. Methods Appl. Sci., 16 (1993), 327-358.  doi: 10.1002/mma.1670160503.

[38]

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), 031504, 18 pp. doi: 10.1063/1.4793988.

[39]

Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys., 60 (2009), 54-69.  doi: 10.1007/s00033-008-6122-6.

[40]

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.

[41]

J. E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 629-648.  doi: 10.1090/qam/1306041.

[42]

J. E. Muñoz Rivera and E. Cabanillas Lapa, Decay rates of solutions of an anisotropic inhomogeneous $n$-dimensional viscoelastic equation with polynomially decaying kernels, Comm. Math. Phys., 177 (1996), 583-602.  doi: 10.1007/BF02099539.

[43]

J. E. Muñoz RiveraE. C. Lapa and R. Barreto, Decay rates for viscoelastic plates with memory, J. Elasticity, 44 (1996), 61-87.  doi: 10.1007/BF00042192.

[44]

J. E. Muñoz Rivera and H. P. Oquendo, Exponential stability to a contact problem of partially viscoelastic materials, J. Elasticity, 63 (2001), 87-111.  doi: 10.1023/A:1014091825772.

[45]

J. E. Muñoz Rivera and A. Peres Salvatierra, Asymptotic behaviour of the energy in partially viscoelastic materials, Quart. Appl. Math., 59 (2001), 557-578.  doi: 10.1090/qam/1848535.

[46]

M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Mathematical Methods in the Applied Sciences, 41 (2018), 192-204.  doi: 10.1002/mma.4604.

[47]

M. I. Mustafa, Energy decay in a quasilinear system with finite and infinite memories, in Mathematical Methods in Engineering, Springer, 2019,235–256.

[48]

N. Najdi and A. Wehbe, Weakly locally thermal stabilization of Bresse systems, Electron. J. Differential Equations, (2014), No. 182, 19 pp.

[49]

W. Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal., 48 (1972), 1-50.  doi: 10.1007/BF00253367.

[50]

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

[51]

M. de L. SantosA. Soufyane and D. da S. Almeida Júnior, Asymptotic behavior to Bresse system with past history, Quart. Appl. Math., 73 (2015), 23-54.  doi: 10.1090/S0033-569X-2014-01382-4.

[52]

M. L. Santos and D. da S. Almeida Júnior, Numerical exponential decay to dissipative Bresse system, J. Appl. Math., 2010, (2010), Art. ID 848620, 17 pp. doi: 10.1155/2010/848620.

[53]

J. A. SorianoW. Charles and R. Schulz, Asymptotic stability for Bresse systems, J. Math. Anal. Appl., 412 (2014), 369-380.  doi: 10.1016/j.jmaa.2013.10.019.

[54]

A. Soufyane and B. Said-Houari, The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system, Evol. Equ. Control Theory, 3 (2014), 713-738.  doi: 10.3934/eect.2014.3.713.

[55]

V. Volterra, Sur les équations intégro-différentielles et leurs applications, Acta Math., 35 (1912), 295-356.  doi: 10.1007/BF02418820.

[56]

C.-C. Wang, The principle of fading memory, Arch. Rational Mech. Anal., 18 (1965), 343-366.  doi: 10.1007/BF00281325.

[57]

A. Wehbe and W. Youssef, Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks, J. Math. Phys., 51 (2010), 103523, 17 pp. doi: 10.1063/1.3486094.

[58]

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

show all references

References:
[1]

M. AfilalA. GuesmiaA. Soufyane and M. Zahri, On the exponential and polynomial stability for a linear Bresse system, Math. Methods Appl. Sci., 43 (2020), 2626-2645.  doi: 10.1002/mma.6070.

[2]

A. M. Al-Mahdi, General stability result for a viscoelastic plate equation with past history and general kernel, J. Math. Anal. Appl., 490 (2020), 124216, 19 pp. doi: 10.1016/j.jmaa.2020.124216.

[3]

A. M. Al-Mahdi, Stability result of a viscoelastic plate equation with past history and a logarithmic nonlinearity, Bound. Value Probl., 2020 (2020), Paper No. 84, 20 pp. doi: 10.1186/s13661-020-01382-9.

[4]

A. M. Al-Mahdi and M. M. Al-Gharabli, New general decay results in an infinite memory viscoelastic problem with nonlinear damping, Bound. Value Probl., 2019 (2019), Paper No. 140, 15 pp. doi: 10.1186/s13661-019-1253-6.

[5]

A. M. Al-MahdiM. M. Al-Gharabli and S. M. Ali, New stability result for a bresse system with one infinite memory in the shear angle equation, Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), 995-1014.  doi: 10.3934/dcdss.2021086.

[6]

A. M. Al-Mahdi, M. M. Al-Gharabli, A. Guesmia and S. A. Messaoudi, New decay results for a viscoelastic-type Timoshenko system with infinite memory, Z. Angew. Math. Phys., 72 (2021), Paper No. 22, 24 pp. doi: 10.1007/s00033-020-01446-x.

[7]

A. M. Al-Mahdi, M. M. Al-Gharabli, M. Kafini and S. Al-Omari, On the global existence and asymptotic behavior of the solution of a nonlinear wave equation with past history, J. Math. Phys., 62 (2021), Paper No. 031512, 15 pp. doi: 10.1063/5.0003515.

[8]

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.

[9]

F. Alabau BoussouiraJ. E. Muñoz Rivera and D. da S. Almeida Júnior, Stability to weak dissipative Bresse system, J. Math. Anal. Appl., 374 (2011), 481-498.  doi: 10.1016/j.jmaa.2010.07.046.

[10]

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

[11]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, vol. 60, Springer-Verlag, New York-Heidelberg, 1978.

[12]

L. Boltzmann, Zur theorie der elastischen nachwirkung, Annalen der Physik, 241 (1878), 430-432. 

[13]

J. A. C. Bresse, Cours de Mécanique Appliquée: Résistance des Matériaux et Stabilitédes Constructions, vol. 1, Mallet-Bachelier, 1859.

[14]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324.  doi: 10.1137/S0363012902408010.

[15]

B. Chentouf, A minimal state approach to dynamic stabilization of the rotating disk-beam system with infinite memory, IEEE Trans. Automat. Control, 61 (2016), 3700-3706.  doi: 10.1109/TAC.2016.2518482.

[16]

B. Chentouf and Z.-J. Han, On the elimination of infinite memory effects on the stability of a nonlinear non-homogeneous rotating body-beam system, Journal of Dynamics and Differential Equations, (2022), 1–25. doi: 10.1007/s10884-021-10111-4.

[17]

B. D. Coleman and V. J. Mizel, Norms and semi-groups in the theory of fading memory, Arch. Rational Mech. Anal., 23 (1966), 87-123.  doi: 10.1007/BF00251727.

[18]

B. D. Coleman and W. Noll, On the thermostatics of continuous media, Arch. Rational Mech. Anal., 4 (1959), 97-128.  doi: 10.1007/BF00281381.

[19]

B. D. Coleman and W. Noll, Foundations of linear viscoelasticity, Rev. Modern Phys., 33 (1961), 239-249.  doi: 10.1103/RevModPhys.33.239.

[20]

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.

[21]

C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations, 7 (1970), 554-569.  doi: 10.1016/0022-0396(70)90101-4.

[22]

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

[23]

G. Dassios and F. Zafiropoulos, Equipartition of energy in linearized $3$-D viscoelasticity, Quart. Appl. Math., 48 (1990), 715-730.  doi: 10.1090/qam/1079915.

[24]

L. H. Fatori and R. N. Monteiro, The optimal decay rate for a weak dissipative Bresse system, Appl. Math. Lett., 25 (2012), 600-604.  doi: 10.1016/j.aml.2011.09.067.

[25]

L. H. Fatori and J. E. M. Rivera, Rates of decay to weak thermoelastic Bresse system, IMA Journal of Applied Mathematics, 75 (2010), 881-904. 

[26]

C. GiorgiJ. E. 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.

[27]

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.

[28]

A. Guesmia, Asymptotic stability of Bresse system with one infinite memory in the longitudinal displacements, Mediterr. J. Math., 14 (2017), Paper No. 49, 19 pp. doi: 10.1007/s00009-017-0877-y.

[29]

A. Guesmia, Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement, Nonauton. Dyn. Syst., 4 (2017), 78-97.  doi: 10.1515/msds-2017-0008.

[30]

A. Guesmia, New general decay rates of solutions for two viscoelastic wave equations with infinite memory, Math. Model. Anal., 25 (2020), 351-373.  doi: 10.3846/mma.2020.10458.

[31]

A. Guesmia and M. Kafini, Bresse system with infinite memories, Math. Methods Appl. Sci., 38 (2015), 2389-2402.  doi: 10.1002/mma.3228.

[32]

A. Guesmia and M. Kirane, Uniform and weak stability of Bresse system with two infinite memories, Z. Angew. Math. Phys., 67 (2016), Art. 124, 39 pp. doi: 10.1007/s00033-016-0719-y.

[33]

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

[34]

A. Guesmia and N.-E. Tatar, Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay, Commun. Pure Appl. Anal., 14 (2015), 457-491.  doi: 10.3934/cpaa.2015.14.457.

[35]

X. Han and M. Wang, General decay estimate of energy for the second order evolution equations with memory, Acta Appl. Math., 110 (2010), 195-207.  doi: 10.1007/s10440-008-9397-x.

[36]

W. J. Hrusa, Global existence and asymptotic stability for a semilinear hyperbolic volterra equation with large initial data, SIAM J. Math. Anal., 16 (1985), 110-134.  doi: 10.1137/0516007.

[37]

J. E. LagneseG. Leugering and E. J. P. G. Schmidt, Modelling of dynamic networks of thin thermoelastic beams, Math. Methods Appl. Sci., 16 (1993), 327-358.  doi: 10.1002/mma.1670160503.

[38]

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), 031504, 18 pp. doi: 10.1063/1.4793988.

[39]

Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys., 60 (2009), 54-69.  doi: 10.1007/s00033-008-6122-6.

[40]

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.

[41]

J. E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 629-648.  doi: 10.1090/qam/1306041.

[42]

J. E. Muñoz Rivera and E. Cabanillas Lapa, Decay rates of solutions of an anisotropic inhomogeneous $n$-dimensional viscoelastic equation with polynomially decaying kernels, Comm. Math. Phys., 177 (1996), 583-602.  doi: 10.1007/BF02099539.

[43]

J. E. Muñoz RiveraE. C. Lapa and R. Barreto, Decay rates for viscoelastic plates with memory, J. Elasticity, 44 (1996), 61-87.  doi: 10.1007/BF00042192.

[44]

J. E. Muñoz Rivera and H. P. Oquendo, Exponential stability to a contact problem of partially viscoelastic materials, J. Elasticity, 63 (2001), 87-111.  doi: 10.1023/A:1014091825772.

[45]

J. E. Muñoz Rivera and A. Peres Salvatierra, Asymptotic behaviour of the energy in partially viscoelastic materials, Quart. Appl. Math., 59 (2001), 557-578.  doi: 10.1090/qam/1848535.

[46]

M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Mathematical Methods in the Applied Sciences, 41 (2018), 192-204.  doi: 10.1002/mma.4604.

[47]

M. I. Mustafa, Energy decay in a quasilinear system with finite and infinite memories, in Mathematical Methods in Engineering, Springer, 2019,235–256.

[48]

N. Najdi and A. Wehbe, Weakly locally thermal stabilization of Bresse systems, Electron. J. Differential Equations, (2014), No. 182, 19 pp.

[49]

W. Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal., 48 (1972), 1-50.  doi: 10.1007/BF00253367.

[50]

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

[51]

M. de L. SantosA. Soufyane and D. da S. Almeida Júnior, Asymptotic behavior to Bresse system with past history, Quart. Appl. Math., 73 (2015), 23-54.  doi: 10.1090/S0033-569X-2014-01382-4.

[52]

M. L. Santos and D. da S. Almeida Júnior, Numerical exponential decay to dissipative Bresse system, J. Appl. Math., 2010, (2010), Art. ID 848620, 17 pp. doi: 10.1155/2010/848620.

[53]

J. A. SorianoW. Charles and R. Schulz, Asymptotic stability for Bresse systems, J. Math. Anal. Appl., 412 (2014), 369-380.  doi: 10.1016/j.jmaa.2013.10.019.

[54]

A. Soufyane and B. Said-Houari, The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system, Evol. Equ. Control Theory, 3 (2014), 713-738.  doi: 10.3934/eect.2014.3.713.

[55]

V. Volterra, Sur les équations intégro-différentielles et leurs applications, Acta Math., 35 (1912), 295-356.  doi: 10.1007/BF02418820.

[56]

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Figure 1.  Test 1: The approximate solution $ (\varphi, \psi, w) $ in the $ x $-$ t $ plane
Figure 2.  Test 1: Energy decay
Figure 3.  Test 2: The approximate solution $ (\varphi, \psi, w) $ in the $ x $-$ t $ plane
Figure 4.  Test 2: Energy decay
Figure 5.  Test 3: The approximate solution $ (\varphi, \psi, w) $ in the $ x $-$ t $ plane
Figure 6.  Test 3: Energy decay
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