Long-time dynamics for a nonlinear Timoshenko system with delay

References

• Timoshenko S. On the correction for shear of the differential equation for transverse vibrations of prismaticbars. Philos. Mag. 1921;41:744–746.
   Google Scholar
• Soufyane A. Stabilisation de la poutre de Timoshenko. C. R. Acad. Sci. Paris Sér. I Math. 1999;328:731–734.
   Google Scholar
• Soufyane A, Wehbe A. Uniform stabilization for the Timoshenko beam by a locally distributed damping. Electron. J. Differ. Equ. 2003;29:14 p.
   Google Scholar
• Almeida Júnior DS, Muñoz Rivera JE, Santos ML. The stability number of the Timoshenko system with second sound. J. Differ. Equ. 2012;253:2715–2733.
   Web of Science ®Google Scholar
• Amar-Khodja F, Benabdallah A, Muñoz Rivera JE, et al. Energy decay for Timoshenko systems of memory type. J. Differ. Equ. 2003;194:82–115.
   Web of Science ®Google Scholar
• Fatori LH, Monteiro RN, Fernández Sare HD. The Timoshenko system with history and Cattaneo law. Appl. Math. Comput. 2014;228:128–140.
   Web of Science ®Google Scholar
• Guesmia A, Messaoudi SA. General energy decay estimates of timoshenko systems with frictional versus viscoelastic damping. Math. Meth. Appl. Sci. 2009;32:2102–2122.
   Web of Science ®Google Scholar
• Ma Z, Zhang L, Yang X. Exponential stability for a Timoshenko-type system with history. J. Math. Anal. Appl. 2011;380:299–312.
   Web of Science ®Google Scholar
• Messaoudi SA, Mustafa MI. On the stabilization of the Timoshenko system by a weak nonlinear dissipation. Math. Methods Appl. Sci. 2009;32:454–469.
   Web of Science ®Google Scholar
• Messaoudi SA, Pokojovy M, Said-Houari B. Nonlinear damped Timoshenko systems with second sound-global existence and exponential stability. Math. Methods Appl. Sci. 2009;32:505–534.
   Web of Science ®Google Scholar
• Muñoz Rivera JE, Racke R. Timoshenko systems with indefinite damping. J. Math. Anal. Appl. 2008;341:1068–1083.
   Web of Science ®Google Scholar
• Muñoz Rivera JE, Racke R. Global stability for damped Timoshenko systems. Discrete Contin. Dyn. Syst. 2003;9:1625–1639.
   Web of Science ®Google Scholar
• Said-Houari B, Laskri Y. A stability result of a Timoshenko system with a delay term in the internal feedback. Appl. Math. Comput. 2010;217:2857–2869.
   Web of Science ®Google Scholar
• Fridman E. Introduction to time-delay systems, Analysis and control. Cham: Birkhäuser/Springer; 2014.
   Google Scholar
• Hale JK, Verduyn SM. Introduction to functional-differential equations. New York (NY): Springer-Verlag; 1993.
   Google Scholar
• Datko R, Lagnese J, Polis MP. An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 1986;24:152–156.
   Web of Science ®Google Scholar
• 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. 2006;45:1561–1585.
   Web of Science ®Google Scholar
• Nicaise S, Pignotti C. Stabilization of the wave equation with boundary or internal distributed delay. Differ. Int. Equ. 2008;21:935–958.
   Web of Science ®Google Scholar
• Feng B, Pelicer ML. Global existence and exponential stability for a nonlinear Timoshenko system with delay. Boundary Value Prob. 2015;2015;1–13. doi:10.1186/s13661-015-0468-4.
   Web of Science ®Google Scholar
• Apalara TA. Well-posedness and exponential stability for a linear damped Timoshenko system with second sound and internal distributed delay. Electron. J. Differ. Equ. 2014;254:15 p.
   Google Scholar
• Apalara TA. Asymptotic behavior of weakly dissipative Timoshenko system with internal constant delay feedbacks. Appl. Anal. 2016;95:187–202.
   Web of Science ®Google Scholar
• Apalara TA, Messaoudi SA. An exponential stability result of a Timoshenko system with thermoelasticity with second sound and in the presence of delay. Appl. Math. Optim. 2015;71:449–472.
   Web of Science ®Google Scholar
• Benaissa A, Bahlil M. Global existence and energy decay of solutions to a nonlinear Timoshenko beam system with a delay term. Taiwanese J. Math. 2014;18:1411–1437.
   Web of Science ®Google Scholar
• Guesmia A. Some well-posedness and general stability results in Timoshenko systems with infinite memory and distributed time delay. J. Math. Phys. 2014;55:081503. 40 p.
   Web of Science ®Google Scholar
• Kafini M, Messaoudi SA, Mustafa MI. Energy decay result in a Timoshenko-type system of thermoelasticity of type III with distributive delay. J. Math. Phys. 2013;54:101503. 14 p.
   Web of Science ®Google Scholar
• Kafini M, Messaoudi SA, Mustafa MI. Energy decay rates for a Timoshenko-type system of thermoelasticity of type III with constant delay. Appl. Anal. 2014;93:1201–1216.
   Web of Science ®Google Scholar
• Kirane M, Said-Houari B, Anwar MN. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Commun. Pure Appl. Anal. 2011;10:667–686.
   Web of Science ®Google Scholar
• Said-Houari B, Soufyane A. Stability result of the Timoshenko system with delay and boundary feedback. IMA J. Math. Control Inform. 2012;29:383–398.
   Web of Science ®Google Scholar
• Xu G, Wang H. Stabilisation of Timoshenko beam system with delay in the boundary control. Internat. J. Control. 2013;86:1165–1178.
   Web of Science ®Google Scholar
• Kafini M, Messaoudi SA, Mustafa MI, et al. Well-posedness and stability results in a Timoshenko-type system of thermoelasticity of type III with delay. Z. Angew. Math. Phys. 2015;66:1499–1517.
   Web of Science ®Google Scholar
• Liu Z, Zheng S. Semigroups associated with dissipative systems. Vol. 398., Chapman Hall & CRC, Research Notes in Mathematics; Boca Raton (FL), USA: Chapman Hall & CRC; 1999.
   Google Scholar
• Pazy A. Semigroups of linear operators and applications to partial diffrential equations. Vol. 44, Applied Mathematical Sciences; New York (NY), USA: Springer; 1983.
   Google Scholar
• Chueshov ID. Introduction to the theory of infinite dimensional dissipative systems. Kharkiv, Ukraine: Acta Scientific Publishing House; 2002.
   Google Scholar
• Chueshov ID, Lasiecka I. Long-time behavior of second order evolution equations with nonlinear damping. Vol. 195, Memoirs of the American Mathematical Society; Providence: American Mathematical Society; 2008.
   Google Scholar
• Chueshov ID, Lasiecka I. Von Karman evolution equations. New York (NY): Springer Verlag; 2012.
   Google Scholar
• Hale JK. Asymptotic behavior of dissipative systems. Vol. 25, Mathematical surveys and monograpgs; Providence: American Mathematical Society; 1988.
   Google Scholar
• Temam R. Infinite-dimensional dynamical systems in mechanics and physics. Vol. 68, Applied mathematical sciences; New York (NY): Springer-Verlag; 1988.
   Google Scholar
• Jorge MA, Ma TF. Long-time dynamics for a class of Kirchhoff models with memory. J. Math. Phys. 2013;54:15 p. Artcile ID: 021505.
   Web of Science ®Google Scholar