Abstract
We consider an operator semigroup arising in the study of an integro-differential equation modeling small transverse vibrations of a viscoelastic pipeline with allowance for Kelvin–Voigt damping. The explicit form of the generator of this semigroup is established, its spectrum is studied, and the norm of its resolvent is estimated. Based on these results, the analyticity of this semigroup is proved with the use of well-known facts from the theory of semigroups.
Similar content being viewed by others
REFERENCES
Pivovarchik, V.N., A boundary value problem related to vibrations of a rod with external and internal damping, Vestn. Mosk. Gos. Univ. Ser. 1. Mat. Mekh., 1987, no. 3, pp. 68–71.
Il’yushin, A.A. and Pobedrya, B.E., Osnovy matematicheskoi teorii termovyazkouprugosti (Basics of the Mathematical Theory of Thermoviscoelasticity), Moscow: Nauka, 1970.
Lancaster, P. and Shkalikov, A., Damped vibrations of beams and related spectral problems, Can. Appl. Math. Quart., 1994, vol. 2, no. 1, pp. 45–90.
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.
Vlasov, V.V., Rautian, N.A., and Shamaev, A.S., Analysis of operator models arising in problems of hereditary mechanics, J. Math. Sci., 2012, vol. 201, no. 5, pp. 673–692.
Vlasov, V.V. and Rautian, N.A., Spectral analysis of integro-differential equations in a Hilbert space, Sovrem. Mat. Fundam. Napravleniya, 2012, vol. 62, pp. 53–71.
Vlasov, V.V. and Rautian, N.A., Spektral’nyi analiz funktsional’no-differentsial’nykh uravnenii (Spectral Analysis of Functional-Differential Equations), Moscow: MAKS Press, 2016.
Pandolfi, L. and Ivanov, S., Heat equations with memory: lack of controllability to the rest, J. Math. Appl., 2009, vol. 355, pp. 1–11.
Pandolfi, L., The controllability of the Gurtin–Pipkin equations: a cosine operator approach, Appl. Math. Optim., 2005, vol. 52, pp. 143–165.
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.
Amendola, G., Fabrizio, M., and Golden, J.M., Thermodynamics of Materials with Memory, Boston: Springer, 2012.
Dafermos, C.M., Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 1970, vol. 37, pp. 297–308.
Fabrizio, M., Giorgi, C., and Pata, V., A new approach to equations with memory, Arch. Ration. Mech. Anal., 2010, vol. 198, pp. 189–232.
Azizov, T.Ya., Kopachevskii, N.D., and Orlova, L.D., An operator approach to the study of the Oldroyd hydrodynamic model, Math. Notes, 1999, vol. 65, no. 6, pp. 773–776.
Zakora, D.A., Exponential stability of a certain semigroup and applications,Math. Notes, 2018, vol. 103, no. 5, pp. 745–760.
Eremenko, A. and Ivanov, S., Spectra of Gurtin–Pipkin type equations,SIAM J. Math. Anal., 2011, vol. 43, pp. 2296–2306.
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.
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.
Miloslavskii, A.I., Instability spectrum of an operator bundle, Math. Notes, 1991, vol. 49, no. 4, pp. 391–395.
Kato, T., Perturbation Theory for Linear Operators, Heidelberg–New York: Springer-Verlag, 1965. Translated under the title:Teoriya vozmushchenii lineinykh operatorov, Moscow: Mir, 1972.
Engel, K.-J. and Nagel, R., One-Parameter Semigroup for Linear Evolution Equations. GTM, New York: Springer-Verlag, 1999.
Tretter, C., Spectral Theory of Block Operator Matrices and Applications, London: Imp. Coll. Press, 2008.
Akhiezer, N.I. and Glazman, I.M., Teoriya lineinykh operatorov v gil’bertovom prostranstve (Theory of Linear Operators in a Hilbert Space), Khar’kov: Izd. Khar’k. Univ., 1977.
Gokhberg, I.Ts. and Krein, M.G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gil’bertovom prostranstve (Introduction to the Theory of Nonself-Adjoint Operators in a Hilbert Space), Moscow: Nauka, 1965.
ACKNOWLEDGMENTS
The author is grateful to Prof. V.V. Vlasov for posing this problem and for constant interest in the research as well as to all participants of the seminar headed by V.V. Vlasov for helpful advice, discussions, and guidance.
Funding
This work was supported by the Center for Fundamental and Applied Mathematics at Lomonosov Moscow State University.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by V. Potapchouck
Rights and permissions
About this article
Cite this article
Tikhonov, Y.A. Analyticity of an Operator Semigroup Arising in Viscoelasticity Problems. Diff Equat 56, 797–812 (2020). https://doi.org/10.1134/S0012266120060129
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266120060129