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Existence and energy decay of a nonuniform Timoshenko system with second sound

  • Taklit Hamadouche
  • Salim A. Messaoudi
Article
  • 120 Downloads

Abstract

In this paper, we consider a linear thermoelastic Timoshenko system with variable physical parameters, where the heat conduction is given by Cattaneo’s law and the coupling is via the displacement equation. We discuss the well-posedness and the regularity of solution using the semigroup theory. Moreover, we establish the exponential decay result provided that the stability function \(\chi _{r}(x)=0\). Otherwise, we show that the solution decays polynomially.

Keywords

Nonuniform Timoshenko system Thermoelasticity Second sound Well-posedness Exponential decay Polynomial decay 

Mathematics Subject Classification

35B37 35L55 74D05 93D45 93D20 

References

  1. 1.
    Ammar-Khodja, F., Benabdallah, A., Muñoz Rivera, J.E., Racke, R.: Energy decay for Timoshenko systems of memory type. J. Differ. Equ. 194(1), 82–115 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ammar-Khodja, F., Kerbal, S., Soufyane, A.E.: Stabilization of the nonuniform Timoshenko beam. J. Math. Anal. Appl. 327(1), 525–538 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Apalara, T.A., Messaoudi, S.A., Keddi, A.: On the decay rates of Timoshenko system with second sound. Math. Methods Appl. Sci. 39(10), 2671–2684 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Almeida Júnior, D.S., Santos, M.L., Muñoz Rivera, J.E.: Stability to 1-D thermoelastic Timoshenko beam acting on shear force. Zeitschrift fr angewandte Mathematik und Physik 65(6), 1233–1249 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fernández Sare, H.D., Racke, R.: On the stability of damped Timoshenko systems: Cattaneo versus Fourier law. Arch. Ration. Mech. Anal 194(1), 221–251 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kim, J.U., Renardy, Y.: Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25(6), 1417–1429 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Messaoudi, S.A., Mustafa, M.I.: A stability result in a memory-type Timoshenko system. Dyn.Syst. Appl. 18, 457–468 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Messaoudi, S.A., Pokojovy, M., Said-Houari, B.: Nonlinear damped Timoshenko systems with second sound-global existence and exponential stability. Math. Methods Appl. Sci. 32(5), 505–534 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Muñoz Rivera, J.E., Racke, R.: Global stability for damped Timoshenko systems. Discrete Contin. Dyn. Syst 9(6), 1625–1639 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Muñoz Rivera, J.E., Racke, R.: Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability. J. Math. Anal. Appl 276, 248–276 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Raposo, C.A., Ferreira, J., Santos, M.L., Castro, N.N.O.: Exponential stability for the Timoshenko system with two weak dampings. Appl. Math. Lett. 18, 535–541 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Santos, M.L., Almeida Junior, D.S., Muñoz Rivera, J.E.: The stability number of the Timoshenko system with second sound. J. Differ. Equ. 253(9), 2715–2733 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Shi, D.H., Feng, D.X.: Exponential decay of Timoshenko beam with locally distributed feedback. IMA J. Math. Control Inf. 18(3), 395–403 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Soufyane, A., Wehbe, A.: Uniform stabilization for the Timoshenko beam by a locally distributed damping. Electron. J. Differ. Equ. 2003(29), 1–14 (2003)MathSciNetGoogle Scholar
  15. 15.
    Taylor, S.W.: Boundary control of a Timoshenko beam with variable physical characteristics. Research Report 356. University of Auckland, Department of Mathematics (1998)Google Scholar
  16. 16.
    Timoshenko, S.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. 41, 744–746 (1921)CrossRefGoogle Scholar
  17. 17.
    Yan, Q., Chen, Z., Feng, D.: Exponential stability of nonuniform Timoshenko beam with coupled locally distributed feedbacks. Acta Anal. Funct. Appl. 5(2), 156–164 (2003)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics, AMNEDP LaboratoryUSTHBEl Alia, Bab Ezzouar, AlgiersAlgeria
  2. 2.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia

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