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Spectral Analysis of Integrodifferential Equations in Hilbert Spaces

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Abstract

We investigate the well-posedness of initial-value problems for abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space and provide the spectral analysis of operator-functions describing symbols of such equations. These equations are an abstract form of linear partial integrodifferential equations arising in the viscoelasticity theory and other important applications. We establish the localization and the spectrum structure of operator-functions describing symbols of these equations.

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References

  1. W. Desch and R.K. Miller, “Exponential stabilization of Volterra integrodifferential equations in Hilbert space,” J. Differ. Equ., 70, 366–389 (1987).

    Article  MATH  Google Scholar 

  2. G. Di Blasio, “Parabolic Volterra equations of convolution type,” J. Integr. Equ. Appl., 6, 479–508 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  3. G. Di Blasio, K. Kunisch, and E. Sinestari, “L 2-regularity for parabolic partial integrodifferential equations with delays in the highest order derivatives,” J. Math. Anal. Appl., 102, 38–57 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Di Blasio, K. Kunisch, and E. Sinestari, “Stability for abstract linear functional differential equations,” Israel J. Math., 50, No. 3, 231–263 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  5. I.Ts. Gokhberg and M. G. Kreyn, Introduction to the Theory of Linear Nonselfadjoint Operators [in Russian], Nauka, Moscow (1965).

  6. M. E. Gurtin and A.C. Pipkin, “Theory of heat conduction with finite wave speed,” Arch. Ration. Mech. Anal., 31, 113–126 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  7. A.A. Il’yushin and B. E. Pobedrya, Essentials of Mathematical Theory of Thermoviscoelasticity [in Russian], Nauka, Moscow (1970).

  8. S. Ivanov and L. Pandolfi, “Heat equations with memory: lack of controllability to the rest,” J. Math. Anal. Appl., 355, 1–11 (2009).

    Article  MathSciNet  MATH  Google Scholar 

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

  10. N.D. Kopachevsky and S.G. Krein, Operator Approach to Linear Problems of Hydrodynamics. Vol. 2. Nonselfadjoint Problems for Viscous Fluids, Birkh¨auser, Berlin–Basel–Boston (2003).

  11. K. Kunisch and M. Mastinsek, “Dual semigroups and structural operators for partial differential equations with unbounded operators acting on the delays,” Differ. Integr. Equ., 3, No. 4, 733–756 (1990).

    MathSciNet  MATH  Google Scholar 

  12. J. Lions and E. Magenes, Nonhomogeneous Boundary-Value Problems and Their Applications [Russian translation], Mir, Moscow (1971).

  13. A. V. Lykov, Problem of Heat-Mass Exchange [in Russian], Nauka i tekhnika, Minsk (1976).

  14. D. A. Medvedev, V.V. Vlasov, and J. Wu, “Solvability and structural properties of abstract neutral functional differential equations,” Funct. Differ. Equ., 66, No. 3-4, 249–272 (2008).

    MathSciNet  MATH  Google Scholar 

  15. R.K. Miller, “Volterra integral equation in Banach space,” Funkcial. Ekvac., 18, 163–194 (1975).

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. R.K. Miller and R. L. Wheeler, “Well-posedness and stability of linear Volterra integrodifferential equations in abstract spaces,” Funkcial. Ekvac., 21, 279–305 (1978).

    MathSciNet  MATH  Google Scholar 

  18. A. I. Miloslavskii, “Spectral properties of operator pencils arising in viscoelasticity,” Deposited in Ukr. NIINTI, No. 1229-87 (1987).

  19. V. V. Palin and E.V. Radkevich, “Conservation laws and their hyperbolic regularizations,” Sovrem. Mat. Prilozh., 64, 80–101 (2009).

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Yu.N. Rabotnov, Elements of Heritable Mechanics of Solid Bodies [in Russian], Nauka, Moscow (1977).

  22. E. Sanches-Palensiya, Nonhomogeneous Media and Oscillation Theory [Russian translation], Mir, Moscow (1984).

  23. A. S. Shamaev and V.V. Shumilova, “Averaging of the acoustics equations for viscoelastic matter with channels filled with viscous compressible fluid,” Izv. Ross. Akad. Nauk Mekh. Zhidk. Gaza, No. 2, 92–103 (2011).

  24. J. Shapiro, Composition Operators and Classical Function Theory, Springer, New York (1993).

    Book  MATH  Google Scholar 

  25. A. A. Shkalikov, “Strongly dumped operator pencils and solvability of corresponding operatordifferential equations,” Mat. Sb., 177, No. 1, 96–118 (1988).

    Google Scholar 

  26. A. A. Shkalikov, “Elliptic equations in a Hilbert space and related spectral problems,” Tr. Semin. im. I.G. Petrovskogo, 14, 140–224 (1989).

    MATH  Google Scholar 

  27. V. V. Vlasov, “On solvability and properties of solutions of functional differential equations in a Hilbert space,” Mat. Sb., 186, No. 8, 67–92 (1995).

    MathSciNet  Google Scholar 

  28. V. V. Vlasov, “On correct solvability of abstract parabolic equations with aftereffect,” Dokl. Akad. Nauk, 415, No. 2, 151-152 (1995).

    Google Scholar 

  29. V. V. Vlasov, “On solvability and estimates of solutions of functional differential equations in Sobolev spaces,” Tr. Mat. Inst. Steklova, 227, 109–121 (1999).

    MathSciNet  MATH  Google Scholar 

  30. V. V. Vlasov, A. A. Gavrikov, S.A. Ivanov, D.Yu. Knyaz’kov, V.A. Samarin, and A. S. Shamaev, “Spectral properties of combined media,” Sovrem. Mat. Prilozh., 64, 105–199 (2009).

    MATH  Google Scholar 

  31. V. V. Vlasov and D. A. Medvedev, “Functional differential equations in Sobolev spaces and related questions of the spectral theory,” Sovrem. Mat. Fundam. Napravl., 30, 3–173 (2008).

    Google Scholar 

  32. V. V. Vlasov, D. A. Medvedev, and N. A. Rautian, Functional Differential Equations in Sobolev Spaces and Their Spectral Analysis [in Russian], MSU, Moscow (2011).

  33. V. V. Vlasov and N. A. Rautian, “Correct solvability and spectral analysis of abstract hyperbolic integrodifferential equations,” Tr. Semin. im. I. G. Petrovskogo, 28, 75–114 (2011).

    MATH  Google Scholar 

  34. V. V. Vlasov and N. A. Rautian, “On properties of solutions of integrodifferential equations arising in the heat-mass exchange theory,” Tr. Mosk. Mat. Obs., 75, No. 2, 131–155 (2014).

    Google Scholar 

  35. V. V. Vlasov and N. A. Rautian, “Correct solvability and spectral analysis of integrodifferential equations arising in the viscoelasticity theory,” Sovrem. Mat. Fundam. Napravl., 58, 22–42 (2015).

    Google Scholar 

  36. V. V. Vlasov and N. A. Rautian, “Correct solvability of Volterra integrodifferential equations in a Hilbert space,” Differ. Uravn., 52, No. 9, 1168–1177 (2016).

    Google Scholar 

  37. V. V. Vlasov, N. A. Rautian, and A.C. Shamaev, “Solvability and spectral analysis of integrodifferential equations arising in thermal physics and acoustics,” Dokl. Akad. Nauk, 434, No. 1, 12–15 (2010).

    Google Scholar 

  38. V. V. Vlasov, N. A. Rautian, and A.C. Shamaev, “Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermal physics and acoustics,” Sovrem. Mat. Fundam. Napravl., 39, 36–65 (2011).

    Google Scholar 

  39. V. V. Vlasov and K. I. Shmatov, “Correct solvability of hyperbolic-type equations with delay in a Hilbert space,” Tr. Mat. Inst. Steklova, 243, 127–137 (2003).

    MATH  Google Scholar 

  40. V. V. Vlasov and J. Wu, “Solvability and spectral analysis of abstract hyperbolic equations with delay,” Funct. Differ. Equ., 16, No. 4, 751–768 (2009).

    MathSciNet  MATH  Google Scholar 

  41. V. V. Vlasov, J. Wu, and G. R. Kabirova, “Correct solvability and spectral properties of abstract hyperbolic equations with aftereffect,” Sovrem. Mat. Fundam. Napravl., 35, 44–59 (2010).

    MathSciNet  MATH  Google Scholar 

  42. J. Wu, “Semigroup and integral form of class of partial differential equations with infinite delay,” Differ. Integr. Equ., 4, No. 6, 1325–1351 (1991).

    MathSciNet  MATH  Google Scholar 

  43. J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York (1996).

    Book  MATH  Google Scholar 

  44. V. V. Zhikov, “On one extension and application of the two-scale convergence method,” Mat. Sb., 186, No. 8, 67–92 (1995).

    MathSciNet  Google Scholar 

  45. V. V. Zhikov, “On two-scale convergence,” Tr. Semin. im. I. G. Petrovskogo, 23, 149–187 (2003).

    Google Scholar 

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Authors and Affiliations

  1. M. V. Lomonosov Moscow State University, Moscow, Russia

    V. V. Vlasov & N. A. Rautian

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  1. V. V. Vlasov
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  2. N. A. Rautian
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Correspondence to V. V. Vlasov.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 62, Differential and Functional Differential Equations, 2016.

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Vlasov, V.V., Rautian, N.A. Spectral Analysis of Integrodifferential Equations in Hilbert Spaces. J Math Sci 239, 771–787 (2019). https://doi.org/10.1007/s10958-019-04325-7

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  • DOI: https://doi.org/10.1007/s10958-019-04325-7

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