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Well-Posedness and Spectral Analysis of Hyperbolic Volterra Equations of Convolution Type

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Differential and Difference Equations with Applications (ICDDEA 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 164))

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Abstract

We study the correct solvability of abstract integrodifferential equations in Hilbert space generalizing integrodifferential equations arising in the theory of viscoelasticity. The equations under considerations are the abstract hyperbolic equations perturbed by the terms containing Volterra integral operators. We establish the correct solvability in the weighted Sobolev spaces of vector-valued functions on the positive semiaxis. We also provide the spectral analysis of operator-valued functions which are the symbols of these equations.

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References

  1. Amendola, G., Fabrizio, M., Golden, J.M.: Thermodynamics of Materials with Memory: Theory and Applications. Springer, New York (2012)

    Book  MATH  Google Scholar 

  2. Desch, W., Miller, R.K.: Exponential stabilization of Volterra Integrodifferential equations in Hilbert space. J. Differ. Equ. 70, 366–389 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Gurtin, M.E., Pipkin, A.C.: General theory of heat conduction with finite wave speed. Arch. Ration. Mech. Anal. 31, 113–126 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1966)

    Book  MATH  Google Scholar 

  5. Kopachevsky, N.D., Krein, S.G.: Operator approach to linear problems of hydrodynamics. In: Nonself Adjoint Problems for Viscous Fluids, vol. 2. Birkhäuser, Berlin/Basel/Boston (2003)

    Google Scholar 

  6. Lions, J.L., Magenes, E.: Nonhomogeneous Boundary-Value Problems and Applications. Springer, Berlin/Heidelberg/New York (1972)

    Book  MATH  Google Scholar 

  7. Lykov, A.V.: Problems of Heat and Mass Transfer. Nauka i Tekhnika, Minsk (1976). (in Russian)

    Google Scholar 

  8. Miller, R.K.: Volterra integral equation in Banach space. Funkcialaj Ekvac. 18, 163–194 (1975)

    MATH  Google Scholar 

  9. Miller, R.K.: An integrodifferential equation for rigid heat conductors with memory. J. Math. Anal. Appl. 66, 313–332 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  10. Pandolfi, L.: The controllability of the Gurtin-Pipkin equations: a cosine operator approach. Appl. Math. Optim. 52, 143–165 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Sanches-Palencia, E. Non-Homogeneous Media and Vibration Theory. Springer, New York (1980)

    Google Scholar 

  12. Shkalikov, A.A.: Strongly damped operator pencil and solvability of according operator differential equations. Sb. Math. 177 (1), 96–118 (1998)

    MathSciNet  Google Scholar 

  13. Vlasov, V.V., Medvedev, D.A.: Functional-differential equations in Sobilev spaces and related problems of spectral theory. Contemp. Math. Fundam. Dir. 30, 3–173 (2008). (English translation J. Math. Sci. 164 (5), 659–841 (2010))

    Google Scholar 

  14. Vlasov, V.V., Rautian, N.A.: Well-defined solvability and spectral analysis of abstract hyperbolic integrodifferential equations. J. Math. Sci. 179 (3), 390–414 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Vlasov, V.V., Gavrikov, A.A., Ivanov, S.A., Knyazkov, D.U., Samarin, V.A., Shamaev, A.S.: Spectral properties of the combined medies. Mod. Probl. Math. Mech. 1, 134–155 (2009)

    Google Scholar 

  16. Vlasov, V.V., Rautian, N.A., Shamaev, A.S.: Solvability and spectral analysis of integro-differential equations arising in the theory of heat transfer and acoustics. Dokl. Math. 82, 684–687 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Vlasov, V.V., Wu, J., Kabirova, G.R.: Well-defined solvability and spectral properties of abstract hyperbolic equations with aftereffect. J. Math. Sci. 170 (3), 388–404 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Vlasov, V.V., Medvedev, D.A., Rautian, N.A.: Functional differential equations in Sobolev spaces and its spectral analysis. In: Contemporary Problems of Mathematics and Mechanics, vol. 8(1), 308 P. Moscow State University, Moscow (2011)

    Google Scholar 

  19. Vlasov, V.V., Rautian, N.A., Shamaev, A.S.: Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics. J. Math. Sci. 190 (1), 34–65 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wu, J.: Theory and Applications of Partial Functional Differential Equations. Applied Mathematical Sciences, vol. 119. Springer, New York (1996)

    Google Scholar 

  21. Zhikov, V.V.: On an extension of the method of two-scale convergence and its applications. Sb. Math. 191 (7), 31–72 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    N. A. Rautian & V. V. Vlasov

Authors
  1. N. A. Rautian
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  2. V. V. Vlasov
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Corresponding author

Correspondence to N. A. Rautian .

Editor information

Editors and Affiliations

  1. Departamento de Ciências Exactas e Naturais, Academia Militar, Amadora, Portugal

    Sandra Pinelas

  2. Department of Mathematics and Statistics, Masaryk University, Brno, Czech Republic

    Zuzana Došlá

  3. Department of Mathematics and Statistics, Masaryk University, Brno, Czech Republic

    Ondřej Došlý

  4. School of Mathematics and Stastistics, Huazhong University of Science and Technology, Wuhan, China

    Peter E. Kloeden

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Rautian, N.A., Vlasov, V.V. (2016). Well-Posedness and Spectral Analysis of Hyperbolic Volterra Equations of Convolution Type. In: Pinelas, S., Došlá, Z., Došlý, O., Kloeden, P. (eds) Differential and Difference Equations with Applications. ICDDEA 2015. Springer Proceedings in Mathematics & Statistics, vol 164. Springer, Cham. https://doi.org/10.1007/978-3-319-32857-7_38

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