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doi: 10.3934/dcdss.2021086
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New stability result for a Bresse system with one infinite memory in the shear angle equation

Adel M. Al-Mahdi 1,2,, , Mohammad M. Al-Gharabli 1,2,3, and  Saeed M. Ali 1,2,3,

1. 

The Preparatory Year Program
2. 

The Interdisciplinary Research Center in Construction and Building Materials, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
3. 

Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam 34151, Saudi Arabia

* Corresponding author: Adel M. Al-Mahdi

Received  March 2021 Revised  May 2021 Early access July 2021

Fund Project: This paper is supported by KFUPM grant #SB191037

In this paper, we consider a one-dimensional linear Bresse system with only one infinite memory acting in the second equation (the shear angle equation) of the system. We prove that the asymptotic stability of the system holds under some general condition imposed into the relaxation function, precisely,
$ g^{\prime}(t)\le -\xi(t) G(g(t)). $
The proof is based on the multiplier method and makes use of convex functions and some inequalities. More specifically, we remove the constraint imposed on the boundedness condition on the initial data
$ \eta{0x} $
. This study generalizes and improves previous literature outcomes.
Keywords: Stability, Bress System, infinite memory, convexity.
Mathematics Subject Classification: Primary: 35B40, 93D15, 35A25; Secondary: 93D20.
Citation: Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Saeed M. Ali. New stability result for a Bresse system with one infinite memory in the shear angle equation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021086
References:
[1]

M. O. Alves, L. H. Fatori, M. A. Jorge Silva and R. N. Monteiro, Stability and optimality of decay rate for a weakly dissipative bresse system, Mathematical Methods in the Applied Sciences, 38 (2015), 898-908.  doi: 10.1002/mma.3115.  Google Scholar

[2]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60, Springer-Verlag, New York-Heidelberg (1978). doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[3]

A. M. Al-Mahdi, General stability result for a viscoelastic plate equation with past history and general kernel, Journal of Mathematical Analysis and Applications, 490 (2020), 124216, 1–19. doi: 10.1016/j.jmaa.2020.124216.  Google Scholar

[4]

A. M. Al-Mahdi, Stability result of a viscoelastic plate equation with past history and a logarithmic nonlinearity, Boundary Value Problems, 2020 (2020), 1-20.  doi: 10.1186/s13661-020-01382-9.  Google Scholar

[5]

F. A. Boussouira, J. E. M. Rivera and D. da S. A. Júnior, Stability to weak dissipative Bresse system, J. Math. Anal. Appl., 374 (2011), 481-498.  doi: 10.1016/j.jmaa.2010.07.046.  Google Scholar

[6]

J. A. Bresse, Cours De Mecanique Appliquee: Re'sistance Des Mate'riaux Et Stabilite'des Constructions, Mallet-Bachelier, Paris (1859). doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[7]

W. Charles, J. A. Soriano, F. A. F. Nascimento and J. H. Rodrigues, Decay rates for bresse system with arbitrary nonlinear localized damping, Journal of Differential Equations, 255 (2013), 2267-2290.  doi: 10.1016/j.jde.2013.06.014.  Google Scholar

[8]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Archive for Rational Mechanics and Analysis, 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[9]

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.  Google Scholar

[10]

L. H. Fatori and J. E. M. Rivera, Rates of decay to weak thermoelastic Bresse system, IMA J. Appl. Math., 75 (2010), 881-904.  doi: 10.1093/imamat/hxq038.  Google Scholar

[11]

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

[12]

A. Guesmia and M. Kirane, Uniform and weak stability of Bresse system with two infinite memories, Z. Angew. Math. Phys., 67 (2016), 1-39.  doi: 10.1007/s00033-016-0719-y.  Google Scholar

[13]

A. Guesmia, Asymptotic stability of Bresse system with one infinite memory in the longitudinal displacements, Mediterr. J. Math., 14 (2017), 1-19.  doi: 10.1007/s00009-017-0877-y.  Google Scholar

[14]

A. Guesmia and S. A. Messaoudi, A general stability result in a Timoshenko system with infinite memory: A new approach, Math. Methods Appl. Sci., 37 (2014), 384-392.  doi: 10.1002/mma.2797.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

J. E. Lagnese, G. 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.  Google Scholar

[18]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, analysis and control of dynamic elastic multi-link structures, in Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA (1994). doi: 10.1007/978-1-4612-0273-8.  Google Scholar

[19]

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

[20]

N. Noun and A. Wehbe, Stabilisation faible interne locale de système élastique de Bresse, C. R. Math. Acad. Sci. Paris, 350 (2012), 493-498.  doi: 10.1016/j.crma.2012.04.003.  Google Scholar

[21]

N. Najdi and A. Wehbe, Weakly locally thermal stabilization of Bresse systems, Electron. J. Differential Equations, 182 (2014), 1-19.   Google Scholar

[22]

M. L. Santos, D. S. A. Júnior and J. E. M. Rivera, The stability number of the Timoshenko system with second sound, J. Differential Equations, 253 (2012), 2715-2733.  doi: 10.1016/j.jde.2012.07.012.  Google Scholar

[23]

J. A. Soriano, J. E. M. Rivera and L. H. Fatori, Bresse system with indefinite damping, J. Math. Anal. Appl., 387 (2012), 284-290.  doi: 10.1016/j.jmaa.2011.08.072.  Google Scholar

[24]

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

[25]

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.  Google Scholar

[26]

J. A. Soriano, W. 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.  Google Scholar

[27]

A. Wehbe and W. Youssef, Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks, J. Math. Phys., 51 (2010), 1067-1078.  doi: 10.1063/1.3486094.  Google Scholar

show all references

References:
[1]

M. O. Alves, L. H. Fatori, M. A. Jorge Silva and R. N. Monteiro, Stability and optimality of decay rate for a weakly dissipative bresse system, Mathematical Methods in the Applied Sciences, 38 (2015), 898-908.  doi: 10.1002/mma.3115.  Google Scholar

[2]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60, Springer-Verlag, New York-Heidelberg (1978). doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[3]

A. M. Al-Mahdi, General stability result for a viscoelastic plate equation with past history and general kernel, Journal of Mathematical Analysis and Applications, 490 (2020), 124216, 1–19. doi: 10.1016/j.jmaa.2020.124216.  Google Scholar

[4]

A. M. Al-Mahdi, Stability result of a viscoelastic plate equation with past history and a logarithmic nonlinearity, Boundary Value Problems, 2020 (2020), 1-20.  doi: 10.1186/s13661-020-01382-9.  Google Scholar

[5]

F. A. Boussouira, J. E. M. Rivera and D. da S. A. Júnior, Stability to weak dissipative Bresse system, J. Math. Anal. Appl., 374 (2011), 481-498.  doi: 10.1016/j.jmaa.2010.07.046.  Google Scholar

[6]

J. A. Bresse, Cours De Mecanique Appliquee: Re'sistance Des Mate'riaux Et Stabilite'des Constructions, Mallet-Bachelier, Paris (1859). doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[7]

W. Charles, J. A. Soriano, F. A. F. Nascimento and J. H. Rodrigues, Decay rates for bresse system with arbitrary nonlinear localized damping, Journal of Differential Equations, 255 (2013), 2267-2290.  doi: 10.1016/j.jde.2013.06.014.  Google Scholar

[8]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Archive for Rational Mechanics and Analysis, 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[9]

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.  Google Scholar

[10]

L. H. Fatori and J. E. M. Rivera, Rates of decay to weak thermoelastic Bresse system, IMA J. Appl. Math., 75 (2010), 881-904.  doi: 10.1093/imamat/hxq038.  Google Scholar

[11]

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

[12]

A. Guesmia and M. Kirane, Uniform and weak stability of Bresse system with two infinite memories, Z. Angew. Math. Phys., 67 (2016), 1-39.  doi: 10.1007/s00033-016-0719-y.  Google Scholar

[13]

A. Guesmia, Asymptotic stability of Bresse system with one infinite memory in the longitudinal displacements, Mediterr. J. Math., 14 (2017), 1-19.  doi: 10.1007/s00009-017-0877-y.  Google Scholar

[14]

A. Guesmia and S. A. Messaoudi, A general stability result in a Timoshenko system with infinite memory: A new approach, Math. Methods Appl. Sci., 37 (2014), 384-392.  doi: 10.1002/mma.2797.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

J. E. Lagnese, G. 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.  Google Scholar

[18]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, analysis and control of dynamic elastic multi-link structures, in Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA (1994). doi: 10.1007/978-1-4612-0273-8.  Google Scholar

[19]

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

[20]

N. Noun and A. Wehbe, Stabilisation faible interne locale de système élastique de Bresse, C. R. Math. Acad. Sci. Paris, 350 (2012), 493-498.  doi: 10.1016/j.crma.2012.04.003.  Google Scholar

[21]

N. Najdi and A. Wehbe, Weakly locally thermal stabilization of Bresse systems, Electron. J. Differential Equations, 182 (2014), 1-19.   Google Scholar

[22]

M. L. Santos, D. S. A. Júnior and J. E. M. Rivera, The stability number of the Timoshenko system with second sound, J. Differential Equations, 253 (2012), 2715-2733.  doi: 10.1016/j.jde.2012.07.012.  Google Scholar

[23]

J. A. Soriano, J. E. M. Rivera and L. H. Fatori, Bresse system with indefinite damping, J. Math. Anal. Appl., 387 (2012), 284-290.  doi: 10.1016/j.jmaa.2011.08.072.  Google Scholar

[24]

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

[25]

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.  Google Scholar

[26]

J. A. Soriano, W. 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.  Google Scholar

[27]

A. Wehbe and W. Youssef, Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks, J. Math. Phys., 51 (2010), 1067-1078.  doi: 10.1063/1.3486094.  Google Scholar

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