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Exponential characterization of a fully damped Timoshenko-Boltzmann system

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Abstract

In this article, we revisit the Timoshenko-Boltzmann system proposed by Liu and Peng [22] and characterize the exponential stability of the associated solution semigroup by considering a more general class of memory kernels. Our main result gives a slight generalization of the stability results achieved in [19, 22].

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References

  1. Arendt, W., Batty, C.J.K.: Tauberian theorems and stability of one-parameter semigroups. Trans. Am. Math. Soc. 306, 837–852 (1988)

    Article  MathSciNet  Google Scholar 

  2. Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-valued laplace transforms and cauchy problems. Birkhäuser, Basel (2011)

  3. Boltzmann, L.: Zur Theorie der elastischen Nachwirkung. Wien. Ber. 70, 275–306 (1874)

    Google Scholar 

  4. Boltzmann, L.: Zur Theorie der elastischen Nachwirkung. Wied. Ann. 5, 430–432 (1878)

    Article  Google Scholar 

  5. Calsavara, B.M.R., Gomes Tavares, E.H., Jorge Silva, M.A.: Exponential stability for a thermo-viscoelastic Timoshenko system with fading memory. J. Math. Anal. Appl. 512, 126147 (2022)

    Article  MathSciNet  Google Scholar 

  6. Chepyzhov, V.V., Pata, V.: Some remarks on stability of semigroups arising from linear viscoelasticity. Asymp. Anal. 46, 251–273 (2006)

    MathSciNet  Google Scholar 

  7. Conti, M., Dell’Oro, F., Pata, V.: Some unexplored questions arising in linear viscoelasticity. J. Func. Anal. 282, 109422 (2022)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Dafermos, C.M.: Asymptotic stability in viscoelasticity. Arch. Rational Mech. Anal. 37, 297–308 (1970)

    Article  MathSciNet  Google Scholar 

  10. Danese, V., Dell’Oro, F., Pata, V.: Stability analysis of abstract systems of Timoshenko type. J. Evol. Equ. 3(16), 587–615 (2016)

    Article  MathSciNet  Google Scholar 

  11. Dell’Oro, F., Pata, V.: Lack of exponential stability in Timoshenko systems with flat memory kernels. Appl. Math. Optim. 71, 79–93 (2015)

    Article  MathSciNet  Google Scholar 

  12. Drozdov, A.D., Kolmanovskii, V.B.: Stability in Viscoelasticity (North-Holland, 1994)

  13. Fabrizio, M., Giorgi, C., Pata, V.: A new approach to equations with memory. Arch. Rational Mech. Anal. 198, 189–232 (2010)

    Article  MathSciNet  Google Scholar 

  14. Folland, G.B.: Real Analysis: Modern Techniques and Their Applications (Wiley, 1999)

  15. Gatti, S., Miranville, A., Pata, V., Zelik, S.: Attractors for semi-linear equations of viscoelasticity with very low dissipation. Rocky Mount. J. Math. 38, 1117–1138 (2008)

    Article  MathSciNet  Google Scholar 

  16. Gearhart, L.: Spectral theory for contraction semigroups on Hilbert spaces. Trans. Amer. Math. Soc. 236, 385–394 (1978)

    Article  MathSciNet  Google Scholar 

  17. Gomes Tavares, E.H., Jorge Silva, M.A., Ma, T.F., Oquendo, H.P.: Shearing viscoelasticity in partially dissipative Timoshenko-Boltzmann systems. SIAM J. Math. Anal. 56(1), 1149–1178 (2024)

    Article  MathSciNet  Google Scholar 

  18. Grasselli, M., Pata, V.: Uniform attractors of nonautonomous dynamical systems with memory. Prog. Nonlin. Differ. Equ. Appl. 50, 155–178 (2002)

    MathSciNet  Google Scholar 

  19. Grasselli, M., Pata, V., Prouse, G.: Longtime behavior of a viscoelastic Timoshenko beam. Discrete Contin. Dyn. Syst. 10, 337 (2004)

    Article  MathSciNet  Google Scholar 

  20. Guesmia, A., Messaoudi, S.A.: A general stability result in a timoshenko system with infinite memory: a new approach. Math. Methods Appl. Sci. 37, 384–392 (2014)

    Article  MathSciNet  Google Scholar 

  21. Lagnese, J.E., Leugering, Günter., Schmidt, E.J.P.G.: Modeling, analysis and control of dynamic elastic multi-link structures, Springer Science and Business Media (2012)

  22. Liu, Z., Peng, C.: Exponential stability of a viscoelastic Timoshenko beam. Adv. Math. Sci. Appl. 8, 343–351 (1998)

    MathSciNet  Google Scholar 

  23. Messaoudi, S.A., Said-Houari, B.: Uniform decay in a Timoshenko-type system with past history. J. Math. Anal. Appl. 360, 459–475 (2009)

    Article  MathSciNet  Google Scholar 

  24. Pata, V.: Exponential stability in linear viscoelasticity. Quart. Appl. Math. 64, 499–513 (2006)

    Article  MathSciNet  Google Scholar 

  25. Pata, V.: Exponential stability in linear viscoelasticity with almost flat memory kernels. Commun. Pure Appl. Anal. 9, 721–730 (2010)

    Article  MathSciNet  Google Scholar 

  26. Pazy, A.: Semigroups of Linear Operator and Applications to Partial Differential Equations Semigroups. Springer-Verlag, New York (1983)

    Book  Google Scholar 

  27. Prüss, J.: Evolutionary integral equations and applications, Monographs in Mathematics, Vol. 87 (Birkhäuser Verlag, 1993)

  28. Prüss, J.: On the spectrum of \(C_0\)-semigroups. Trans. Am. Math. Soc. 284, 847–857 (1984)

    Google Scholar 

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

    Google Scholar 

  30. Timoshenko, S.P.: Vibration Problems in Engineering (Van Nostrand, 1955)

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Funding

M. Barbosa da Silva is a Ph.D. student with scholarship supported by CAPES.

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Correspondence to Eduardo H. Gomes Tavares.

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da Silva, M.B., Delatorre, L.G., Tavares, E.H.G. et al. Exponential characterization of a fully damped Timoshenko-Boltzmann system. Z. Angew. Math. Phys. 75, 79 (2024). https://doi.org/10.1007/s00033-024-02214-x

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  • DOI: https://doi.org/10.1007/s00033-024-02214-x

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