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

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.

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  • 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.