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Existence and energy decay of solution to a nonlinear viscoelastic two-dimensional beam with a delay

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Abstract

A longitudinal and transversal vibrations of the beam with nonlinear tension, a viscoelastic damping and distributed delay term is studied. Using the Faedo–Galerkin method, the well-posedness of the problem is established. A uniform decay result is proved by multiplier method.

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References

  • Aili, M., & Khemmoudj, A. (2019). General decay of energy for a viscoelastic wave equation with a distributed delay term in the nonlinear internal dambing. Rendiconti del Circolo Matematico di Palermo Series, 2, 1–21.

    MATH  Google Scholar 

  • Alabau-Boussouira, F., Cannarsa, P., & Sforza, D. (2008). Decay estimates for second order evolution equations with memory. Journal of Functional Analysis, 254(5), 1342–1372.

    Article  MathSciNet  Google Scholar 

  • Apalara, T. A. (2014). Well-posedness and exponential stability for a linear damped timoshenko system with second sound and internal distributed delay. Electronic Journal of Differential Equations, 254, 15.

    MathSciNet  MATH  Google Scholar 

  • Aubin, J.-P. (1963). Analyse mathematique-un theoreme de compacite. Comptes Rendus Hebdomadaires Des Seances De L Academie Des Sciences, 256(24), 5042.

    MATH  Google Scholar 

  • Benaissa, A., Benguessoum, A., & Messaoudi, S. A. (2014). Global existence and energy decay of solutions to a viscoelastic wave equation with a delay term in the non-linear internal feedback. International Journal of Dynamical Systems and Differential Equations, 5(1), 1–26.

    Article  MathSciNet  Google Scholar 

  • Bland, D. R. (2016). The theory of linear viscoelasticity. Mineola: Courier Dover Publications.

    MATH  Google Scholar 

  • Coleman, B. D., & Noll, W. (1961). Foundations of linear viscoelasticity. Reviews of Modern Physics, 33(2), 239.

    Article  MathSciNet  Google Scholar 

  • Dai, Q., & Yang, Z. (2014). Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay. Zeitschrift für angewandte Mathematik und Physik, 65(5), 885–903.

    Article  MathSciNet  Google Scholar 

  • Do, K. D., & Pan, J. (2008). Boundary control of transverse motion of marine risers with actuator dynamics. Journal of Sound and Vibration, 318(4–5), 768–791.

    Article  Google Scholar 

  • Feng, B. (2015). Global well-posedness and stability for a viscoelastic plate equation with a time delay. Mathematical Problems in Engineering, 2015, 1–10.

    MathSciNet  Google Scholar 

  • Feng, B. (2016). Well-posedness and exponential stability for a plate equation with time-varying delay and past history. Zeitschrift für angewandte Mathematik und Physik, 1(68), 1–24.

    Google Scholar 

  • Feng, B. (2018). General decay for a viscoelastic wave equation with density and time delay term in \(\mathbb{R}^{n}\). Taiwanese Journal of Mathematics, 22(1), 205–223.

    Article  MathSciNet  Google Scholar 

  • Kang, J.-R., Lee, M. J., & Park, S. H. (2017). Asymptotic stability of a viscoelastic problem with Balakrishnan–Taylor damping and time-varying delay. Computers & Mathematics with Applications, 74(6), 1506–1515.

    Article  MathSciNet  Google Scholar 

  • Kelleche, A., & Tatar, N.-E. (2017). Existence and stabilization of a Kirchhoff moving string with a distributed delay in the boundary feedback. Mathematical Modelling of Natural Phenomena, 12(6), 106–117.

    Article  MathSciNet  Google Scholar 

  • Khemmoudj, A., & Mokhtari, Y. (2019). General decay of the solution to a nonlinear viscoelastic modified von-Karman system with delay. Discrete & Continuous Dynamical Systems-A, 39(7), 3839.

    Article  MathSciNet  Google Scholar 

  • Lekdim, B., & Khemmoudj, A. (2018). General decay of energy to a nonlinear viscoelastic two-dimensional beam. Applied Mathematics and Mechanics, 39(11), 1661–1678.

    Article  MathSciNet  Google Scholar 

  • Lions, J. (1969). Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod.

    MATH  Google Scholar 

  • Liu, W., & Sun, Y. (2014). General decay of solutions for a weak viscoelastic equation with acoustic boundary conditions. Zeitschrift für angewandte Mathematik und Physik, 65(1), 125–134.

    Article  MathSciNet  Google Scholar 

  • Liu, W., & Yu, J. (2011). Global existence and uniform decay of solutions for a coupled system of nonlinear viscoelastic wave equations with not necessarily differentiable relaxation functions. Studies in Applied Mathematics, 127(4), 315–344.

    Article  MathSciNet  Google Scholar 

  • Liu, W., Zhu, B., Li, G., & Wang, D. (2017). General decay for a viscoelastic Kirchhoff equation with Balakrishnan–Taylor damping, dynamic boundary conditions and a time-varying delay term. Evolution Equations & Control Theory, 6(2), 239.

    Article  MathSciNet  Google Scholar 

  • Messaoudi, S. A. (2008a). General decay of solutions of a viscoelastic equation. Journal of Mathematical Analysis and Applications, 341(2), 1457–1467.

    Article  MathSciNet  Google Scholar 

  • Messaoudi, S. A. (2008b). General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Analysis: Theory, Methods & Applications, 69(8), 2589–2598.

    Article  MathSciNet  Google Scholar 

  • Mustafa, M. I., & Kafini, M. (2016). Energy decay for viscoelastic plates with distributed delay and source term. Zeitschrift fur Angewandte Mathematik und Physik, 67(3), 36.

    Article  MathSciNet  Google Scholar 

  • Nicaise, S., & Pignotti, C. (2006). Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM Journal on Control and Optimization, 45(5), 1561–1585.

    Article  MathSciNet  Google Scholar 

  • Nicaise, S., & Pignotti, C. (2011). Interior feedback stabilization of wave equations with time dependent delay. Electronic Journal of Differential Equations, 2011, 1–20.

    MathSciNet  MATH  Google Scholar 

  • Nicaise, S., Pignotti, C., et al. (2008). Stabilization of the wave equation with boundary or internal distributed delay. Differential and Integral Equations, 21(9–10), 935–958.

    MathSciNet  MATH  Google Scholar 

  • Park, S. H. (2014). Decay rate estimates for a weak viscoelastic beam equation with time-varying delay. Applied Mathematics Letters, 31, 46–51.

    Article  MathSciNet  Google Scholar 

  • Shinskey, F. G. (1979). Process control systems. New York: McGraw-Hill.

    Google Scholar 

  • Wu, S.-T., et al. (2013). Asymptotic behavior for a viscoelastic wave equation with a delay term. Taiwanese Journal of Mathematics, 17(3), 765–784.

    Article  MathSciNet  Google Scholar 

  • Xu, G. Q., Yung, S. P., & Li, L. K. (2006). Stabilization of wave systems with input delay in the boundary control. ESAIM: Control, Optimisation and Calculus of Variations, 12(4), 770–785.

    MathSciNet  MATH  Google Scholar 

  • Yang, X.-G., Zhang, J., & Wang, S. (2020). Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay. Discrete & Continuous Dynamical Systems-A, 40(3), 1493.

    Article  MathSciNet  Google Scholar 

  • Yang, Z. (2013). On an extensible beam equation with nonlinear damping and source terms. Journal of Differential Equations, 254(9), 3903–3927.

    Article  MathSciNet  Google Scholar 

  • Yang, Z. (2015). Existence and energy decay of solutions for the Euler–Bernoulli viscoelastic equation with a delay. Zeitschrift fur Angewandte Mathematik und Physik, 66(3), 727–745.

    Article  MathSciNet  Google Scholar 

  • Zheng, S. (2004). Nonlinear evolution equations. Boca Raton: CRC Press.

    Book  Google Scholar 

  • Zhong, Q.-C. (2006). Robust control of time-delay systems. Cham: Springer Science & Business Media.

    MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions, and express their gratitude to DGRSDT for the financial support.

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Authors and Affiliations

  1. Department of Mathematics, University Ziane Achour of Djelfa, 17000, Djelfa, Algeria

    Billal Lekdim

  2. Laboratory of SDG, Faculty of Mathematics, University of Science and Technology Houari Boumediene, P.O. Box 32, El-Alia, 16111, Bab Ezzouar, Algiers, Algeria

    Billal Lekdim & Ammar Khemmoudj

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  1. Billal Lekdim
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  2. Ammar Khemmoudj
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Correspondence to Billal Lekdim.

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Lekdim, B., Khemmoudj, A. Existence and energy decay of solution to a nonlinear viscoelastic two-dimensional beam with a delay. Multidim Syst Sign Process 32, 915–931 (2021). https://doi.org/10.1007/s11045-021-00766-z

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  • DOI: https://doi.org/10.1007/s11045-021-00766-z

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