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Exponential decay of thermo-elastic Bresse system with distributed delay term

Year 2018, Volume: 47 Issue: 5, 1216 - 1230, 16.10.2018

Abstract

The paper considered here is one-dimensional linear thermo-elastic Bresse system with a distributed delay term in the first equation. We prove the well-posedness and exponential stability result, this later will be shown without the usual assumption on the wave speeds. To achieve our goals, we make use of the semi-group method.

References

  • Alabau-Boussouira, F., Munôz Rivera, J. E., Almeida Junior, D. S., Stability to weak dissipative Bresse system, Journal of Mathematical Analysis and Applications, 374:481-498, 2011.
  • Benaissa, A., Miloudi, M., Mokhtari, M., Global existence and energy decay of solutions to a Bresse system with delay terms, Comment. Math. Univ. Carolin. 56, 2, 169-186, 2015.
  • Bouzettouta, L., Zitouni, S., Zennir, Kh., and Guesmia, A., Stability of Bresse system with internal distributed delay, J. Math. Comput. Sci. 7, No. 1, 92-118, 2017.
  • Bouzettouta, L., Zitouni, S., Zennir, Kh., and Sissaoui, H., Well-posedness and decay of solutions to Bresse system with internal distributed delay, Int. J. Appl. Math. Stat. 56, 4, 1-12, 2017.
  • Bresse, J. A. C., Cours de Méchanique Appliquée, Mallet Bachelier, Paris, 1859.
  • Feng, B. and Yang, Xin-Guang, Long-time dynamics for a nonlinear Timoshenko system with delay, Applicable Analysis (2016) http://dx.doi.org/10.1080/00036811.2016.1148139.
  • Guesmia, A., Asymptotic stability of abstract dissipative systems with infinite memory, J. Math. Anal. Appl. 382 :748-760, 2011.
  • Guesmia, A. and Kafini, M., Bresse system with infinite memories, Math. Meth. Appl. 2014, DOI: 10.1002/mma.3228.
  • Kim, J. U., Renardy, Y., Boundary control of the Timoshenko beam, SIAM J. Control Optim. 25, 1417-1429, 1987.
  • Liu, Z., Rao, B., Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys. 60, 54-69, 2009.
  • Liu, Z., Zheng, S., Semigroups Associated with Dissipative Systems, 398, Chapman Hall/CRC, London, 1999.
  • Messaoudi, S.A., Mustapha, M.I., On the internal and boundary stabilization of Timoshenko beams, Nonlinear Differ. Equ. Appl. 15, 655-671, 2008.
  • Messaoudi, S.A., Mustapha, M.I., On the stabilization of the Timochenko system by a weak nonlinear dissipation, Math. Meth. Appl. Sci. 32, 454-469, 2009.
  • Munoz Rivera, J.E., Racke, R., Global stability for damped Timoshenko systems, Discrete Contin. Dyn. Syst. Ser. B 9, 16251639, 2003.
  • Nicaise, A. S., Pignotti, C., Stabilization of the wave equation with boundary or internal distributed delay, Dif. Int. Equs. 21 (9-10), 935-958, 2008.
  • Park, J. H. & Kang, J. R., Energy decay of solutions for Timoshenko beam with a weak non-linear dissipation, IMA J. Appl. Math. 76, 340-350, 2011.
  • Pazy, A., Semigroups of linear operators and applications to partial differential equations, Volume 44 of Applied Math. Sciences, Springer-Verlag, New York, 1983.
  • Raposo, C. A., Ferreira, J., Santos, J., Castro, N.N.O., Exponential stability for the Timoshenko system with two weak dampings, Appl. Math. Lett. 18, no. 5, 535-541, 2005.
  • Santos, M. L., Soufyane, A. and Almeida Junior, D. S., Asymptotic behavior to Bresse system with past history, Quarterly Of Applied Mathematics. V LXXIII, 1 23-54, 2015.
  • Timoshenko, S. P.,On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philos. Magazine 6, 744-746, 1921.
  • Zitouni, S., Ardjouni, A., Zennir, Kh. and Amiar, R., Existence and stability of solution for transmission system with varying delay, Int. J. Appl. Math. Stat. 55; Issue No. 2; 2016, 1-13.
Year 2018, Volume: 47 Issue: 5, 1216 - 1230, 16.10.2018

Abstract

References

  • Alabau-Boussouira, F., Munôz Rivera, J. E., Almeida Junior, D. S., Stability to weak dissipative Bresse system, Journal of Mathematical Analysis and Applications, 374:481-498, 2011.
  • Benaissa, A., Miloudi, M., Mokhtari, M., Global existence and energy decay of solutions to a Bresse system with delay terms, Comment. Math. Univ. Carolin. 56, 2, 169-186, 2015.
  • Bouzettouta, L., Zitouni, S., Zennir, Kh., and Guesmia, A., Stability of Bresse system with internal distributed delay, J. Math. Comput. Sci. 7, No. 1, 92-118, 2017.
  • Bouzettouta, L., Zitouni, S., Zennir, Kh., and Sissaoui, H., Well-posedness and decay of solutions to Bresse system with internal distributed delay, Int. J. Appl. Math. Stat. 56, 4, 1-12, 2017.
  • Bresse, J. A. C., Cours de Méchanique Appliquée, Mallet Bachelier, Paris, 1859.
  • Feng, B. and Yang, Xin-Guang, Long-time dynamics for a nonlinear Timoshenko system with delay, Applicable Analysis (2016) http://dx.doi.org/10.1080/00036811.2016.1148139.
  • Guesmia, A., Asymptotic stability of abstract dissipative systems with infinite memory, J. Math. Anal. Appl. 382 :748-760, 2011.
  • Guesmia, A. and Kafini, M., Bresse system with infinite memories, Math. Meth. Appl. 2014, DOI: 10.1002/mma.3228.
  • Kim, J. U., Renardy, Y., Boundary control of the Timoshenko beam, SIAM J. Control Optim. 25, 1417-1429, 1987.
  • Liu, Z., Rao, B., Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys. 60, 54-69, 2009.
  • Liu, Z., Zheng, S., Semigroups Associated with Dissipative Systems, 398, Chapman Hall/CRC, London, 1999.
  • Messaoudi, S.A., Mustapha, M.I., On the internal and boundary stabilization of Timoshenko beams, Nonlinear Differ. Equ. Appl. 15, 655-671, 2008.
  • Messaoudi, S.A., Mustapha, M.I., On the stabilization of the Timochenko system by a weak nonlinear dissipation, Math. Meth. Appl. Sci. 32, 454-469, 2009.
  • Munoz Rivera, J.E., Racke, R., Global stability for damped Timoshenko systems, Discrete Contin. Dyn. Syst. Ser. B 9, 16251639, 2003.
  • Nicaise, A. S., Pignotti, C., Stabilization of the wave equation with boundary or internal distributed delay, Dif. Int. Equs. 21 (9-10), 935-958, 2008.
  • Park, J. H. & Kang, J. R., Energy decay of solutions for Timoshenko beam with a weak non-linear dissipation, IMA J. Appl. Math. 76, 340-350, 2011.
  • Pazy, A., Semigroups of linear operators and applications to partial differential equations, Volume 44 of Applied Math. Sciences, Springer-Verlag, New York, 1983.
  • Raposo, C. A., Ferreira, J., Santos, J., Castro, N.N.O., Exponential stability for the Timoshenko system with two weak dampings, Appl. Math. Lett. 18, no. 5, 535-541, 2005.
  • Santos, M. L., Soufyane, A. and Almeida Junior, D. S., Asymptotic behavior to Bresse system with past history, Quarterly Of Applied Mathematics. V LXXIII, 1 23-54, 2015.
  • Timoshenko, S. P.,On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philos. Magazine 6, 744-746, 1921.
  • Zitouni, S., Ardjouni, A., Zennir, Kh. and Amiar, R., Existence and stability of solution for transmission system with varying delay, Int. J. Appl. Math. Stat. 55; Issue No. 2; 2016, 1-13.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

S. Zitouni This is me

L. Bouzettouta This is me

Kh. Zennir This is me

D. Ouchenane This is me

Publication Date October 16, 2018
Published in Issue Year 2018 Volume: 47 Issue: 5

Cite

APA Zitouni, S., Bouzettouta, L., Zennir, K., Ouchenane, D. (2018). Exponential decay of thermo-elastic Bresse system with distributed delay term. Hacettepe Journal of Mathematics and Statistics, 47(5), 1216-1230.
AMA Zitouni S, Bouzettouta L, Zennir K, Ouchenane D. Exponential decay of thermo-elastic Bresse system with distributed delay term. Hacettepe Journal of Mathematics and Statistics. October 2018;47(5):1216-1230.
Chicago Zitouni, S., L. Bouzettouta, Kh. Zennir, and D. Ouchenane. “Exponential Decay of Thermo-Elastic Bresse System With Distributed Delay Term”. Hacettepe Journal of Mathematics and Statistics 47, no. 5 (October 2018): 1216-30.
EndNote Zitouni S, Bouzettouta L, Zennir K, Ouchenane D (October 1, 2018) Exponential decay of thermo-elastic Bresse system with distributed delay term. Hacettepe Journal of Mathematics and Statistics 47 5 1216–1230.
IEEE S. Zitouni, L. Bouzettouta, K. Zennir, and D. Ouchenane, “Exponential decay of thermo-elastic Bresse system with distributed delay term”, Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 5, pp. 1216–1230, 2018.
ISNAD Zitouni, S. et al. “Exponential Decay of Thermo-Elastic Bresse System With Distributed Delay Term”. Hacettepe Journal of Mathematics and Statistics 47/5 (October 2018), 1216-1230.
JAMA Zitouni S, Bouzettouta L, Zennir K, Ouchenane D. Exponential decay of thermo-elastic Bresse system with distributed delay term. Hacettepe Journal of Mathematics and Statistics. 2018;47:1216–1230.
MLA Zitouni, S. et al. “Exponential Decay of Thermo-Elastic Bresse System With Distributed Delay Term”. Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 5, 2018, pp. 1216-30.
Vancouver Zitouni S, Bouzettouta L, Zennir K, Ouchenane D. Exponential decay of thermo-elastic Bresse system with distributed delay term. Hacettepe Journal of Mathematics and Statistics. 2018;47(5):1216-30.