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New general and explicit stability result for a thermoelastic Timoshenko system

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Abstract

The goal of this article is to study a thermoelastic Timoshenko system with viscoelastic law acting on the shear force and thermoelastic dissipation acting on the bending moment. We prove a general decay as well as an explicit decay estimate for solution energy, from which the exponential and polynomial decay results are only special cases. The result is obtained without imposing equal-wave-speed condition, this is achieved by considering the Neumann–Dirichlet–Neumann boundary conditions . This result is therefore of great interest compared to majority of results in the literature that often time impose the equal-wave-speed condition.

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References

  1. Alves, M.O., et al.: On the modeling and uniform stability of a partially dissipative viscoelastic Timoshenko system. SIAM J. Math. Anal. 51(6), 4520–4543 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ammar-Khodja, F., Benabdallah, A., Muñoz Rivera, J.E., Racke, R.: Energy decay for Timoshenko systems of memory type. J. Differ. Equ. 194, 82–115 (2003)

  3. Apalara, T.A.: General stability of memory-type thermoelastic Timoshenko beam acting on shear force. Continuum Mech. Thermodyn. 30, 291–300 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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(3), 449–472 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Apalara, T.A., Messaoudi, S.A., Keddi, A.A.: On the decay rates of Timoshenko system with second sound. Math. Methods Appl. Sci. 39(10), 2671–2684 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)

    Book  Google Scholar 

  7. Chen, M.M., Liu, W.J., Zhou, W.C.: Existence and general stabilization of the Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms. Adv. Nonlinear Anal. (2018). https://doi.org/10.1515/anona-2016-0085

    Article  MathSciNet  MATH  Google Scholar 

  8. Conti, M., Dell’Oro, F., Pata, V., et al.: Timoshenko systems with fading memory. Dyn. Partial Differ. Equ. 10, 367–377 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Enyi, C.D., Feng, B.: Stability result for a new viscoelastic-thermoelastic Timoshenko system. Bull. Malays. Math. Sci. Soc. (2020). https://doi.org/10.1007/s40840-020-01035-1

    Article  MATH  Google Scholar 

  10. Feng, B.: Uniform decay of energy for a porous thermoelasticity system with past history. Appl. Anal. (2016). https://doi.org/10.1080/00036811.2016.1258116

    Article  Google Scholar 

  11. 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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications, vol. 1. Springer, Berlin (1972)

    Book  MATH  Google Scholar 

  13. Messaoudi, S.A., Fareh, A.: General decay for a porous-thermoelastic system with memory: the case of non equal speeds. Acta Math. Sci. Ser. B Engl. Ed. 33(1), 23–40 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mustafa, M.I.: General decay result for nonlinear viscoelastic equations. J. Math. Anal. Appl. 457, 134–152 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Rivera, J.E.Munoz, Racke, R.: Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability. J. Math. Anal. Appl. 276(1), 248–278 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Timoshenko, S.P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Phil. Mag. 41, 744–746 (1921)

    Article  Google Scholar 

  17. Timoshenko, S.P.: Vibration Problems in Engineering. Van Nostrand, New York (1955)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to appreciate the continuous support of University of Hafr Al Batin.

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Authors and Affiliations

  1. College of Sciences, Department of Mathematics, University of Hafr Al Batin, Saudi Arabia, Hafr Al Batin, 31991, Saudi Arabia

    Cyril Dennis Enyi & Soh Edwin Mukiawa

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  1. Cyril Dennis Enyi
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  2. Soh Edwin Mukiawa
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Correspondence to Soh Edwin Mukiawa.

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Enyi, C.D., Mukiawa, S.E. New general and explicit stability result for a thermoelastic Timoshenko system. Ricerche mat 71, 735–755 (2022). https://doi.org/10.1007/s11587-021-00568-2

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  • DOI: https://doi.org/10.1007/s11587-021-00568-2

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