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On the Properties of a Semigroup of Operators Generated by a Volterra Integro-Differential Equation Arising in the Theory of Viscoelasticity

  • INTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS
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Abstract

Without taking into account external friction, small transverse vibrations of a viscoelastic pipeline of unit length are described for nonnegative values of time in dimensionless variables by an integro-differential equation with hinged conditions at the ends and with initial conditions. The solution of this equation can be written in terms of an operator semigroup. In the present paper, we establish that this equation generates a semigroup that is analytic in some sector of the right half-plane.

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Notes

  1. Here and in the following, by \( \overline {T}\) we denote the closure of an operator \(T \).

REFERENCES

  1. Pivovarchik, V.N., A boundary value problem associated with vibrations of a rod with internal and external friction, Vestn. Mosk. Gos. Univ. Ser. 1. Mat. Mekh., 1987, no. 3, pp. 68–71.

  2. Il’yushin, A.A. and Pobedrya, B.E., Osnovy matematicheskoi teorii termovyazkouprugosti (Fundamentals of the Mathematical Theory of Thermoviscoelasticity), Moscow: Gosizd. Tekh.-Teor. Lit., 1970.

    Google Scholar 

  3. Pipkin, A.C. and Gurtin, M.E., A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal., 1968, vol. 31, pp. 113–126.

    Article  MathSciNet  Google Scholar 

  4. Vlasov, V.V., Rautian, N.A., and Shamaev, A.S., Analysis of operator models arising in problems of hereditary mechanics, J. Math. Sci., 2014, vol. 201, pp. 673–692.

    Article  MathSciNet  Google Scholar 

  5. Vlasov, V.V. and Rautian, N.A., Spectral analysis of integro-differential equations in a Hilbert space, Sovrem. Mat. Fundam. Napravl., 2012, vol. 62, pp. 53–71.

    Google Scholar 

  6. Vlasov, V.V. and Rautian, N.A., Properties of solutions of integro-differential equations arising in heat and mass transfer theory, Trans. Moscow Math. Soc., 2014, vol. 75, pp. 185–204.

    Article  MathSciNet  Google Scholar 

  7. Vlasov, V.V. and Rautian, N.A., Spectral analysis of functional differential equations (Spectral Analysis of Functional Differential Equations), Moscow: Mosk. Gos. Univ., 2016.

    MATH  Google Scholar 

  8. Vlasov, V.V. and Rautian, N.A., Spectral analysis of integro-differential equations in a Hilbert space, Sovrem. Mat. Fundam. Napravl., 2016, vol. 62, pp. 53–71.

    Google Scholar 

  9. Vlasov, V.V. and Rautian, N.A., Well-posedness and spectral analysis of integrodifferential equations arising in viscoelasticity theory, Sovrem. Mat. Fundam. Napravl., 2015, vol. 58, pp. 22–42.

    Google Scholar 

  10. Vlasov, V.V. and Rautian, N.A., Spectral analysis and representation of solutions of integro-differential equations with fractional exponential kernels, Trans. Moscow Math. Soc., 2019, vol. 80, pp. 169–188.

    Article  MathSciNet  Google Scholar 

  11. Rautian, N.A., Semigroups generated by Volterra integro-differential equations, Differ. Equations, 2020, vol. 56, no. 9, pp. 1193–1211.

    Article  MathSciNet  Google Scholar 

  12. Pandolfi, L. and Ivanov, S., Heat equations with memory: lack of controllability to the rest, J. Math. Appl., 2009, vol. 355, pp. 1–11.

    MathSciNet  MATH  Google Scholar 

  13. Pandolfi, L., The controllability of the Gurtin–Pipkin equations: a cosine operator approach, Appl. Math. Optim., 2005, vol. 52, pp. 143–165.

    Article  MathSciNet  Google Scholar 

  14. Rivera, J.E.M. and Naso, M.G., On the decay of the energy for systems with memory and indefinite dissipation, Asympt. Anal., 2006, vol. 49, pp. 189–204.

    MathSciNet  MATH  Google Scholar 

  15. Amendola, G., Fabrizio, M., and Golden, J.M., Thermodynamics of Materials with Memory, Boston: Springer, 2012.

    Book  Google Scholar 

  16. Dafermos, C.M., Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 1970, vol. 37, pp. 297–308.

  17. Fabrizio, M., Giorgi, C., and Pata, V., A new approach to equations with memory, Arch. Ration. Mech. Anal., 2010, vol. 198, pp. 189–232.

    Article  MathSciNet  Google Scholar 

  18. Zakora, D.A., Exponential stability of a certain semigroup and applications, Math. Notes, 2018, vol. 103, no. 5, pp. 745–760.

    Article  MathSciNet  Google Scholar 

  19. Zakora, D.A., Oldroyd’s compressible fluid model, Tr. Krymskoi osennei mat. shkoly-simpoziuma (Proc. Crimean Autumn Math. School-Symp.), Sovrem. Mat. Fundam. Napravl., 2016, vol. 61. pp. 41–66.

  20. Zakora, D.A., Maxwell’s compressible fluid model, Tr. Krymskoi osennei mat. shkoly-simpoziuma (Proc. Crimean Autumn Math. School-Symp.), Sovrem. Mat. Fundam. Napravl., 2017, vol. 63, no. 2, pp. 247–265.

  21. Davydov, A.V., Asymptotics of the spectrum of an integro-differential equation arising in the study of the flutter of a viscoelastic plate, Russ. J. Math. Phys., 2021, vol. 28, no. 2, pp. 188–197.

    Article  MathSciNet  Google Scholar 

  22. Davydov, A.V., Spectral analysis of integrodifferential operators arising in the study of flutter of a viscoelastic plate, Moscow Univ. Math. Bull., 2020, vol. 75, no. 2, pp. 65–71.

    Article  MathSciNet  Google Scholar 

  23. Eremenko, A. and Ivanov, S., Spectra of the Gurtin–Pipkin type equations, SIAM J. Math. Anal., 2011, vol. 43, no. 5, pp. 2296–2306.

    Article  MathSciNet  Google Scholar 

  24. Davydov, A.V. and Tikhonov, Yu.A., Study of Kelvin–Voigt models arising in viscoelasticity, Differ. Equations, 2018, vol. 54, no. 12, pp. 1620–1635.

    Article  MathSciNet  Google Scholar 

  25. Lancaster, P. and Shkalikov, A., Damped vibrations of beams and related spectral problems, Can. Appl. Math. Q., 1994, vol. 2, no. 1, pp. 45–90.

    MathSciNet  MATH  Google Scholar 

  26. Shkalikov, A.A. and Griniv, R.O., On an operator pencil arising in the problem of beam oscillation with internal damping, Math. Notes, 1994, vol. 56, no. 2, pp. 840–851.

    Article  MathSciNet  Google Scholar 

  27. Miloslavskii, A.I., Instability spectrum of an operator bundle, Math. Notes, 1991, vol. 49, no. 4, pp. 391–395.

  28. Kato, T., Perturbation Theory for Linear Operators, Berlin–Heidelberg–New York: Springer, 1966. Translated under the title: Teoriya vozmushchenii lineinykh operatorov, Moscow: Mir, 1972.

    Book  Google Scholar 

  29. Lions, J.-L. and Magenes, E., Problèmes aux limites inhomogènes et applications, Paris: Dunod, 1968.

    MATH  Google Scholar 

  30. Gel’fand, I.M. and Vilenkin, M.Ya., Obobshchennye funktsii (Generalized Functions), Moscow: Gos. Izd. Fiz.-Mat. Lit., 1961.

  31. Akhiezer, N.I. and Glazman, I.M., Teoriya lineinykh operatorov v gil’bertovom prostranstve (Theory of Linear Operators in Hilbert Space), Khar’kov: Izd. Khar’kov. Univ., Vyshcha Shkola, 1977.

  32. Tikhonov, Yu.A., Analyticity of an operator semigroup arising in viscoelasticity problems, Differ. Equations, 2020, vol. 56, no. 6, pp. 797–812.

    Article  MathSciNet  Google Scholar 

  33. Engel, K.-J. and Nagel, R., One-Parameter Semigroup for Linear Evolution Equations, New York: Springer, 1999.

    Google Scholar 

  34. Carathéodory, C., Conformal Representation, Cambridge: Cambridge Univ. Press, 1932.

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ACKNOWLEDGMENTS

The author expresses his deep gratitude to Prof. V.V. Vlasov for posing the problem and constant attention to the work, as well as to all the participants of the seminar under his leadership for useful discussions and valuable advice.

Funding

The study was financially supported by the Interdisciplinary Scientific and Educational School of Lomonosov Moscow State University “Mathematical Methods for the Analysis of Complex Systems.”

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    Yu. A. Tikhonov

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991, Russia

    Yu. A. Tikhonov

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  1. Yu. A. Tikhonov
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Correspondence to Yu. A. Tikhonov.

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Translated by V. Potapchouck

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Tikhonov, Y.A. On the Properties of a Semigroup of Operators Generated by a Volterra Integro-Differential Equation Arising in the Theory of Viscoelasticity. Diff Equat 58, 662–679 (2022). https://doi.org/10.1134/S0012266122050068

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  • DOI: https://doi.org/10.1134/S0012266122050068

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