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Well-posedness and direct internal stability of coupled non-degenrate Kirchhoff system via heat conduction
New stability result for a Bresse system with one infinite memory in the shear angle equation
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 |
$ g^{\prime}(t)\le -\xi(t) G(g(t)). $ |
$ \eta{0x} $ |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
C. M. Dafermos,
Asymptotic stability in viscoelasticity, Archive for Rational Mechanics and Analysis, 37 (1970), 297-308.
doi: 10.1007/BF00251609. |
[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. |
[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. |
[11] |
A. Guesmia and M. Kafini,
Bresse system with infinite memories, Math. Methods Appl. Sci., 38 (2015), 2389-2402.
doi: 10.1002/mma.3228. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
N. Najdi and A. Wehbe,
Weakly locally thermal stabilization of Bresse systems, Electron. J. Differential Equations, 182 (2014), 1-19.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
C. M. Dafermos,
Asymptotic stability in viscoelasticity, Archive for Rational Mechanics and Analysis, 37 (1970), 297-308.
doi: 10.1007/BF00251609. |
[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. |
[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. |
[11] |
A. Guesmia and M. Kafini,
Bresse system with infinite memories, Math. Methods Appl. Sci., 38 (2015), 2389-2402.
doi: 10.1002/mma.3228. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
N. Najdi and A. Wehbe,
Weakly locally thermal stabilization of Bresse systems, Electron. J. Differential Equations, 182 (2014), 1-19.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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