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Cauchy problem for one class of pseudodifferential systems with smooth symbols

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Nonlinear Oscillations

Abstract

For one class of pseudodifferential systems with smooth time-dependent symbols, we study properties of the fundamental matrix of solutions. We formulate sufficient and, for some systems, necessary conditions for the correct solvability of the Cauchy problem with generalized initial data. We also construct spaces of test and generalized functions that generalize certain classical spaces.

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References

  1. I. M. Gel’fand and G. E. Shilov, Spaces of Test and Generalized Functions [in Russian], Fizmatgiz, Moscow (1958).

    Google Scholar 

  2. I. M. Gel’fand and G. E. Shilov, Some Problems of the Theory of Differential Equations [in Russian], Fizmatgiz, Moscow (1958).

    Google Scholar 

  3. V. V. Horodets’kyi, Limit Properties of Solutions of Parabolic Equations Smooth in a Layer [in Ukrainian], Ruta, Chernivtsi (1998).

    Google Scholar 

  4. B. L. Gurevich, “New types of spaces of test and generalized functions and Cauchy problem for systems of finite-difference equations,” Dokl. Akad. Nauk SSSR, 99, No. 6, 893–895 (1954).

    MATH  MathSciNet  Google Scholar 

  5. B. L. Gurevich, “New types of spaces of test and generalized functions and Cauchy problem for differential-difference equations,” Dokl. Akad. Nauk SSSR, 108, No. 6, 1001–1003 (1956).

    MATH  MathSciNet  Google Scholar 

  6. V. A. Litovchenko, “Correct solvability of the Cauchy problem for a pseudodifferential equation of integral type in S-type spaces” Nelin. Gran. Zad., Issue 13, 105–113 ( 2003).

    Google Scholar 

  7. M. Nagase, “On the Cauchy problem for parabolic pseudodifferential equations,” Osaka J. Math., 11, No. 2, 239–264 (1974).

    MATH  MathSciNet  Google Scholar 

  8. R. Shinkai, “On symbols of fundamental solutions of parabolic systems,” Proc. Jpn. Acad., 50, No. 5–6, 337–341 (1974).

    MATH  MathSciNet  Google Scholar 

  9. C. Tsutsumi, “The fundamental solutions for a degenerate parabolic pseudodifferential operator,” Proc. Jpn. Acad., 50, No. 1, 11–15 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  10. C. Tsutsumi, “The fundamental solutions for a parabolic pseudodifferential operator and parametrices for degenerate operators,” Proc. Jpn. Acad., 51, No. 2, 103–108 (1975).

    MATH  MathSciNet  Google Scholar 

  11. L. Schwartz, “Théorie des distributions,” Acra. Sci. Industr., 1, No. 1091 (1950).

  12. V. A. Litovchenko, “Correct solvability of the Cauchy problem for one equation of integral form,” Ukr. Mat. Zh., 56, No. 2, 185–197 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  13. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Their Applications [in Russian], Nauka i Tekhnika, Minsk (1987).

    MATH  Google Scholar 

  14. F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1968).

    Google Scholar 

  15. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs (1964).

  16. V. M. Borok, “Solution of the Cauchy problem for some types of systems of linear partial differential equations,” Dokl. Akad. Nauk SSSR, 97, No 6, 949–952 (1954).

    MATH  MathSciNet  Google Scholar 

  17. V. A. Litovchenko, “Bessel fractional integrodifferentiation with positive parameter,” Nauk. Visn. Cherniv. Univ., Ser. Mat., Issue 134, 65–70 (2002).

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Authors and Affiliations

  1. Chernivtsi National University, Chernivtsi

    V. A. Litovchenko

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  1. V. A. Litovchenko
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Translated from Neliniini Kolyvannya, Vol. 9, No. 4, pp. 502–524, October–December, 2006.

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Litovchenko, V.A. Cauchy problem for one class of pseudodifferential systems with smooth symbols. Nonlinear Oscill 9, 490–512 (2006). https://doi.org/10.1007/s11072-006-0057-7

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  • DOI: https://doi.org/10.1007/s11072-006-0057-7

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