Abstract
In this paper, we consider a one-dimensional linear thermoelastic system of Timoshenko type with delay, where the heat conduction is given by Green and Naghdi’s theory. We establish the well-posedness and the stability of the system for the cases of equal and nonequal speeds of wave propagation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Timoshenko S.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. 41, 744–746 (1921)
Kim J.U., Renardy Y.: Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25(6), 1417–1429 (1987)
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)
Soufyane A., Wehbe A.: Uniform stabilization for the Timoshenko beam by a locally distributed damping. Electron. J. Differ. Equ. 2003(29), 1–14 (2003)
Muñoz Rivera J.E., Racke R.: Global stability for damped Timoshenko systems. Discret. Contin. Dyn. Syst. 9(6), 1625–1639 (2003)
Muñoz Rivera J.E., Racke R.: Timoshenko systems with indefinite damping. J. Math. Anal. Appl. 341, 1068–1083 (2008)
Mustafa M.I., Messaoudi S.A.: General energy decay rates for a weakly damped Timoshenko system. J. Dyn. Control Syst. 16, 211–226 (2010)
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)
Guesmia A., Messaoudi S.A.: General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping. Math. Methods Appl. Sci. 32, 2102–2122 (2009)
Fernández Sare H.D., Muñnoz Rivera J.E.: Stability of Timoshenko systems with past history. J. Math. Anal. Appl. 339 # 1, 482–502 (2008)
Muñoz Rivera J.E., Racke R.: Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability. J. Math. Anal. Appl. 276, 248–276 (2002)
Green A.E., Naghdi P.M.: A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. A. 432, 171–194 (1991)
Green A.E., Naghdi P.M.: On undamped heat waves in an elastic solid. J. Therm. Stress. 15, 253–264 (1992)
Green A.E., Naghdi P.M.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–208 (1993)
Chandrasekharaiah D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51, 705–729 (1998)
Messaoudi S.A., Said-Houari B.: Energy decay in a Timoshenko-type system of thermoelasticity of type III. JMAA 348, 298–307 (2008)
Messaoudi S.A., Fareh A.: Energy decay in a Timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds. Arab. J. Math. 2, 199–207 (2013)
Kafini M.: General energy decay in a Timoshenko-type system of thermoelasticity of type III with a viscoelastic damping. J. Math. Anal. Appl. 375, 523–537 (2011)
Suh I.H., Bien Z.: Use of time delay action in the controller design. IEEE Trans. Automat. Control. 25, 600–603 (1980)
Datko R.: Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Control Optim. 26(3), 697–713 (1988)
Zuazua E.: Exponential decay for the semi-linear wave equation with locally distributed damping. Comm. Partial Differ. Equ. 15, 205–235 (1990)
Nicaise S., Pignotti C.: Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45(5), 1561–1585 (2006)
Ait Benhassi E.M., Ammari K., Boulite S., Maniar L.: Feedback stabilization of a class of evolution equations with delay. J. Evol. Equ. 9, 103–121 (2009)
Nicaise S., Pignotti C., Valein J.: Exponential stability of the wave equation with boundary time-varying delay. Discret. Contin. Dyn. Syst. 2(3), 559–581 (2009)
Racke R.: Instability of coupled systems with delay. Commun. Pure Appl. Anal. 11(5), 1753–1773 (2012)
Guesmia A., Messaoudi S., Soufyane A.: Stabilization of a linear Timoshenko system with infinite history and applications to the Timoshenko-heat systems. Electron. J. Differ. Equ. 193, 1–45 (2012)
Guesmia A.: Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay. IMA J. Math. Control Inf. 30, 507–526 (2013)
Said-Houari B., Soufyane A.: Stability result of the Timoshenko system with delay and boundary feedback. IMA J. Math. Control Inform. 29(3), 383–398 (2012)
Kirane M., Said-Houari B., Anwar M.: Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Commun. Pure Appl. Anal. 10(2), 667–686 (2011)
Said-Houari B., Laskri Y.: A stability result of a Timoshenko system with a delay term in the internal feedback. Appl. Math. Comput. 217(6), 2857–2869 (2010)
Brezis H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kafini, M., Messaoudi, S.A., Mustafa, M.I. et al. Well-posedness and stability results in a Timoshenko-type system of thermoelasticity of type III with delay. Z. Angew. Math. Phys. 66, 1499–1517 (2015). https://doi.org/10.1007/s00033-014-0475-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-014-0475-9