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Dynamics of the Nonlinear Timoshenko System with Variable Delay

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Abstract

This paper is concerned with the wellposedness of global solution and existence of global attractor to the nonlinear Timoshenko system subject to continuous variable time delay in the angular rotation of the beam filament. The waves are assumed to propagate under the same speed in the transversal and angular direction. A single mechanical damping is implemented to counter the destabilizing effect from the time delay term. By imposing appropriate assumptions on the damping term and sub-linear time delay term, we prove the existence of absorbing set and establish the quasi-stability of the gradient system generated from the solution to the system of equation. The quasi-stability property in turn implies the existence of finite dimensional global and exponential attractors that contain the unstable manifold formed from the set of equilibria.

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References

  1. Almeida Júnior, D.S., Mũnoz Rivera, J.E., Santos, M.L.: The stability number of the Timoshenko system with second sound. J. Differ. Equ. 253, 2715–2733 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Almeida Júnior, D.S., Santos, M.L., Mũnoz Rivera, J.E.: Stability to weakly dissipative Timoshenko systems. Math. Meth. Appl. Sci. 36, 1965–1976 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Ammar-Khodja, F., Benabdallah, A., Mũnoz Rivera, J.E.: Energy decay for Timoshenko systems of memory type. J. Differ. Equ. 194, 82–115 (2013)

    MathSciNet  MATH  Google Scholar 

  4. Ammar-Khodja, F., Kerbal, S., Soufyane, A.: Stabilization of the non-uniform Timoshenko beam. J. Math. Appl. Anal. 327(1), 82–115 (2007)

    MATH  Google Scholar 

  5. Apalara, T.A.: Well-posedness and exponential stability for a linear damped Timoshenko system with second sound and internal distributed delay. Electron. J. Differ. Equ. 254, 15 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Apalara, T.A.: Asymptotic behavior of weakly dissipative Timoshenko system with internal constant delay feedbacks. Appl. Anal. 95, 187–202 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Apalara, T.A., Messaoudi, S.A.: An exponential stability result of a Timoshenko system with thermoelasticity with second sound and in the presence of delay. Appl. Math. Optim. 71, 449–472 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Benaissa, A., Bahlil, M.: Global existence and energy decay of solutions to a nonlinear Timoshenko beam system with a delay term. Taiwan. J. Math. 18, 1411–1437 (2014)

    MathSciNet  MATH  Google Scholar 

  9. Chueshov, I.D., Lasiecka, I.: Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping. Memoirs of the American Mathematical Society, vol. 195. American Mathematical Society, Providence (2008)

    MATH  Google Scholar 

  10. Chueshov, I., Lasiecka, I.: Von Karman Evolution Equations. Well-Posedness and Long Time Dynamics. Springer Monographs in Mathematics. Springer, New York (2010)

    MATH  Google Scholar 

  11. Datko, R., Lagnese, J., Polis, M.P.: An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 24, 152–156 (1986)

    MathSciNet  MATH  Google Scholar 

  12. Fatori, L.H., Monteiro, R.N., Fernandez Sare, H.D.: The Timoshenko system with history and Cattaneo law. Appl. Math. Comput. 228, 128–140 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Feng, B., Pelicer, M.L.: Global existence and exponential stability for a nonlinear Timoshenko system with delay. Bound. Value Prob. 2015, 1–13 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Feng, B., Yang, X.: Long-time dynamics for a nonlinear Timoshenko system with delay. Appl. Anal. 96(4), 606–625 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Fernandez Sare, H.D.: Exponential decay of Timoshenko system with indefinite memory dissipation. Adv. Differ. Equ. 13(7–8), 733–752 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Fernandez 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)

    MathSciNet  MATH  Google Scholar 

  17. Fridman, E.: Introduction to Time-Delay Systems, Analysis and Control. Birkhäser/Springer, Cham (2014)

    MATH  Google Scholar 

  18. Grasselli, M., Pata, V., Prouse, G.: Long-time behavior of a viscoelastic Timoshenko beam. Discret. Contin. Dyn. Syst. 10(1–2), 337–348 (2014)

    MATH  Google Scholar 

  19. Guesmia, A.: Some well-posedness and general stability results in Timoshenko systems with infinite memory and distributed time delay. J. Math. Phys. 55, 081503 (2014)

    MathSciNet  MATH  Google Scholar 

  20. 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)

    MathSciNet  MATH  Google Scholar 

  21. Kafini, M., Messaoudi, S.A., Mustafa, M.I.: Energy decay result in a Timoshenko-type system of thermoelasticity of type III with distributive delay. J. Math. Phys. 54, 101503 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Kafini, M., Messaoudi, S.A., Mustafa, M.I.: Energy decay rates for a Timoshenko-type system of thermoelasticity of type III with constant delay. Appl. Anal. 93, 1201–1216 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Kafini, M., Messaoudi, S.A., Mustafa, M.I.: 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)

    MathSciNet  MATH  Google Scholar 

  24. Kim, J., Renardy, Y.: Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25(6), 1417–1429 (1987)

    MathSciNet  MATH  Google Scholar 

  25. Kirane, M., Said-Houari, B., Anwar, M.N.: Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Commun. Pure Appl. Anal. 10, 667–686 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Liu, Z., Zheng, S.: Semigroups Associated with Dissipative Systems. Chapman Hall & CRC, Research Notes in Mathematics, vol. 398. Chapman Hall & CRC, Boca Raton, FL (1999)

    MATH  Google Scholar 

  27. Ma, Z., Zhang, L., Yang, X.: Exponential stability for a Timoshenko-type system with history. J. Math. Anal. Appl. 380, 299–312 (2011)

    MathSciNet  MATH  Google Scholar 

  28. Marín-Rubio, P., Real, J.: Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators. Discret. Contin. Dyn. Syst. 26, 989–1006 (2010)

    MathSciNet  MATH  Google Scholar 

  29. Messaoudi, S.A., Mustafa, M.I.: On the stabilization of the Timoshenko system by a weak nonlinear dissipation. Math. Methods Appl. Sci. 32, 454–469 (2009)

    MathSciNet  MATH  Google Scholar 

  30. 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, 505–534 (2009)

    MathSciNet  MATH  Google Scholar 

  31. Mũnoz Rivera, J.E., Fernandex Sare, H.D.: Stability of Timoshenko systems with past history. J. Math. Anal. Appl. 339(1), 482–502 (2008)

    MathSciNet  MATH  Google Scholar 

  32. Mũnoz Rivera, J.E., Racke, R.: Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability. J. Math. Anal. Appl. 276, 248–278 (2002)

    MathSciNet  MATH  Google Scholar 

  33. Mũnoz Rivera, J.E., Racke, R.: Global stability for damped Timoshenko systems. Discret. Contin. Dyn. Syst. 9, 1625–1639 (2003)

    MathSciNet  MATH  Google Scholar 

  34. Mũnoz Rivera, J.E., Racke, R.: Timoshenko systems with indefinite damping. J. Math. Anal. Appl. 341, 1068–1083 (2008)

    MathSciNet  MATH  Google Scholar 

  35. Mustafa, M.I., Messaoudi, S.A.: General energy decay rates for a weakly damped Timoshenko system. J. Dyn. Control Syst. 16(2), 211–226 (2010)

    MathSciNet  MATH  Google Scholar 

  36. 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, 1561–1585 (2006)

    MathSciNet  MATH  Google Scholar 

  37. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)

    MATH  Google Scholar 

  38. 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)

    MathSciNet  MATH  Google Scholar 

  39. Said-Houari, B., Laskri, Y.: A stability result of a Timoshenko system with a delay term in the internal feedback. Appl. Math. Comput. 217, 2857–2869 (2010)

    MathSciNet  MATH  Google Scholar 

  40. Santos, M.L., Almeida Júnior, D.S., Mũnoz Rivera, J.E.: The stability number of the Timoshenko system with second sound. J. Differ. Equ. 253, 2715–2733 (2013)

    MathSciNet  MATH  Google Scholar 

  41. Said-Houari, B., Soufyane, A.: Stability result of the Timoshenko system with delay and boundary feedback. IMA J. Math. Control Inform. 29, 383–398 (2012)

    MathSciNet  MATH  Google Scholar 

  42. Shi, D.H., Feng, D.X.: Exponential decay of Timoshenko beam with distributed feedback. In: Proceeding of the IFAC Word Congress, Beijing, vol. F (1999)

  43. Soufyane, A.: Stabilisation de la poutre de Timoshenko. C. R. Acad. Sci. Paris Sér I Math. 328, 731–734 (1999)

    MathSciNet  MATH  Google Scholar 

  44. Soufyane, A.: Exponential stability of the linearized non-uniform Timoshenko beam. Nonlinear Anal. Real World Appl. 10, 1016–1020 (2009)

    MathSciNet  MATH  Google Scholar 

  45. Soufyane, A., Wehbe, A.: Uniform stabilization for the Timoshenko beam by a locally distributed damping. Electron. J. Differ. Equ. 29, 1–14 (2003)

    MathSciNet  Google Scholar 

  46. Timoshenko, S.: On the correction for shear of the differential equation for transverse vibrations of prismaticbars. Philos. Mag. 41, 744–746 (1921)

    Google Scholar 

  47. Xu, G., Wang, H.: Stabilisation of Timoshenko beam system with delay in the boundary control. Int. J. Control 86, 1165–1178 (2013)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank the referees’ thorough comments and suggestions that greatly improve the original manuscript. Xin-Guang Yang was partially supported by NSFC of China (Grant No. 11726626), Yongjin Lu was partially supported by NSF of USA (Grant No. 1601127), they were also supported by the Key Project of Science and Technology of Henan Province in China (Grant No. 182102410069).

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Authors and Affiliations

  1. Department of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, People’s Republic of China

    Xin-Guang Yang

  2. Department of Mathematics and Economics, Virginia State University, Petersburg, VA, 23806, USA

    Jing Zhang & Yongjin Lu

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  1. Xin-Guang Yang
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Correspondence to Jing Zhang.

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Yang, XG., Zhang, J. & Lu, Y. Dynamics of the Nonlinear Timoshenko System with Variable Delay. Appl Math Optim 83, 297–326 (2021). https://doi.org/10.1007/s00245-018-9539-0

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