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Energy Decay for von Kármán-Gurtin-Pipkin System

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Abstract

This paper aims to prove the asymptotic behavior of the solution for the thermo-elastic von Karman system where the thermal conduction is given by Gurtin-Pipkins law. Existence and uniqueness of the solution are proved within the semigroup framework and stability is achieved thanks to a suitable Lyapunov functional. Therefore, the stability result clarified that the solutions energy functional decays exponentially at infinite time.

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References

  1. Ackleh, Azmy S., Harvey, Thomas Banks, Gabriella, A. Pinter. A nonlinear beam equation. Applied mathematics letters, 15: 381–387 (2002)

    Article  MathSciNet  Google Scholar 

  2. Ball, J.M. Initial-boundary value problems for an extensible beam. Journal of Mathematical Analysis and Applications, 42: 61–90 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  3. Benabdallah, Assia, Teniou, Djamel. Exponential stability of a Von Karman model with thermal effects. Southwest Texas State University, Department of Mathematics, 1998

  4. Bisognin, E., et al. On exponential stability for Von Karman equations in the presence of thermal effects. Mathematical methods in the applied sciences, 21: 393–416 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cazenave, Thierry, Alain, Haraux. Introduction aux problèmes d’évolution semi-linéaires, Vol.1. Ellipses, 1990

  6. West, H.H. The Euler-Bernoulli beam equation with boundary energy dissipation. Operator Methods for Optimal Control Problems, CRC Press, 67 (1987)

  7. Djebabla, Abdelhak, Nasser-eddine, Tatar. Exponential stabilization of the full von Kármán beam by a thermal effect and a frictional damping. Georgian Mathematical Journal, 20: 427–438 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Favini, Angelo, et al. Global existence, uniqueness and regularity of solutions to a von Karman system with nonlinear boundary dissipation. Differential and Integral Equations, 9: 267–294 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Green, Albert Edward, Paul, M. Naghdi. A re-examination of the basic postulates of thermomechanics. Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, 432: 171–194 (1991)

    MathSciNet  MATH  Google Scholar 

  10. Green, A.E., Naghdi, P.M. Thermoelasticity without energy dissipation. Journal of elasticity, 31: 189–208 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gurtin, Morton E., Allen, C. Pipkin. A general theory of heat conduction with finite wave speeds. Archive for Rational Mechanics and Analysis, 31: 113–126 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lagnese, J.E., Leugering, G. Uniform stabilization of a nonlinear beam by nonlinear boundary feedback. Journal of Differential Equations, 91: 355–388 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lagnese, John E. Boundary stabilization of thin plates. Society for industrial and Applied Mathematics, 1989

  14. Liu, Wenjun, Kewang, Chen, Jun, Yu. Asymptotic stability for a non-autonomous full von Kármán beam with thermo-viscoelastic damping. Applicable Analysis, 97: 400–414 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Menzala, G. Perla, Enrique Zuazua. Timoshenko’s beam equation as limit of a nonlinear one-dimensional von Karman system. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 130: 855–875 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Menzala, Gustavo Perla, Enrique, Zuazua. Explicit exponential decay rates for solutions of von Karmans system of thermoelastic plates. Comptes Rendus de l’Academie des Sciences-Serie I-Mathematique, 324: 49–54 (1997)

    MATH  Google Scholar 

  17. Pazy, Amnon. Semigroups of linear operators and applications to partial differential equations. Vol. 44. Springer Science & Business Media, 2012

  18. Rivera, Jaime E. Muñoz, Reinhard Racke. Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability. Journal of Mathematical Analysis and Applications, 276: 248–278 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Timoshenko, Stephen P.LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 41: 744–746 (1921)

    Article  Google Scholar 

  20. Tucsnak, Marius. Semi-internal Stabilization for a Non-linear Bernoulli-Euler Equation. Mathematical Methods in the Applied Sciences, 19: 897–907 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author would like to thank the editor and the referees for their very helpful suggestions.

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Authors and Affiliations

  1. Laboratory of Applied Mathematics, University of Badji Mokhtar, P.O. Box 12, 23000, Annaba, Algeria

    Hanni Dridi

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  1. Hanni Dridi
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Correspondence to Hanni Dridi.

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Dridi, H. Energy Decay for von Kármán-Gurtin-Pipkin System. Acta Math. Appl. Sin. Engl. Ser. 39, 306–319 (2023). https://doi.org/10.1007/s10255-023-1045-8

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  • DOI: https://doi.org/10.1007/s10255-023-1045-8

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