Abstract
For the Gurtin–Pipkin integro-differential equation whose relaxation kernel can be represented as a Stieltjes integral of a decaying exponential along the positive half-line, we describe a representation of the nonreal spectrum asymptotics versus the asymptotic characteristics of the Stieltjes measure and the behavior of the relaxation kernel itself at zero. The application of the results obtained to the kernels most widely used in practice is demonstrated.
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Notes
Here and in what follows, by \(V_a^b\) we denote the variation of a function on the interval \([a,b]\) .
REFERENCES
Algazin, S.D. and Kiiko, I.A., Flatter plastin i obolochek (Flutter of Plates and Shells), Moscow: Nauka, 2006.
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.
Zhikov, V.V., On an extension of the method of two-scale convergence and its applications, Sb. Math., 2000, no. 7, pp. 973–1014.
Vlasov, V.V. and Rautian, N.A., Well-defined solvability and spectral analysis of abstract hyperbolic integrodifferential equations, J. Math. Sci. (New York), 2011, vol. 179, no. 3, pp. 390–414.
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.
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.
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, Asymptotic Anal., 2006, vol. 49, pp. 189–204.
Amendola, G., Fabrizio, M., and Golden, J.M., Thermodynamics of Materials with Memory. Theory and Applications, New York–Dordrecht–Heidelberg–London: Springer-Verlag, 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.
Vlasov, V.V. and Rautian, N.A., On the properties of semigroups generated by Volterra integro-differential equations, Differ. Equations, 2020, vol. 56, no. 8, pp. 1122–1126.
Tikhonov, Yu.A., Analyticity of an operator semigroup arising in viscoelasticity problems, Differ. Equations, 2020, vol. 56, no. 6, pp. 797–812.
Vlasov, V.V. and Rautian, N.A., Well-posedness and spectral analysis of integrodifferential equations of hereditary mechanics, Comput. Math. Math. Phys., 2020, vol. 60, no. 8, pp. 1322–1330.
Eremenko, A. and Ivanov, S., Spectra of the Gurtin–Pipkin type equations, SIAM J. Math. Anal., 2011, no. 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.
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.
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.
Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of the Theory of Functions and Functional Analysis), Moscow: Nauka, 2012.
Evgrafov, M.A., Sbornik zadach po teorii analiticheskikh funktsii (Collection of Problems on the Theory of Analytic Functions), Moscow: Nauka, 1972.
ACKNOWLEDGMENTS
The author expresses his deep gratitude to Prof. V.V. Vlasov for stating the problem, scientific guidance, and active support, as well as to the participants of the scientific seminar “Functional-differential and integro-differential equations and their spectral analysis,” in particular, N.A. Rautian, for support and valuable advice.
The author is grateful to the referee for constructive remarks that contributed to the improvement of the text of the article.
Funding
This work was supported by a grant from Lomonosov Moscow State University Scientific School “Mathematical Methods for the Analysis of Complex Systems,” led by Acad. V.A. Sadovnichii, as well as with the financial support of the Foundation for the Development of Theoretical Physics and Mathematics “Basis.”
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Translated by V. Potapchouck
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Davydov, A.V. On the Asymptotics of the Nonreal Spectrum of the Integro-Differential Gurtin–Pipkin Equation with Relaxation Kernels Representable in the Form of the Stielties Integral. Diff Equat 58, 242–255 (2022). https://doi.org/10.1134/S0012266122020094
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DOI: https://doi.org/10.1134/S0012266122020094