Advertisement

Ukrainian Mathematical Journal

, Volume 61, Issue 8, pp 1264–1288 | Cite as

Cauchy problem for one class of degenerate parabolic equations of Kolmogorov type with positive genus

  • S. D. Ivasyshen
  • V. A. Litovchenko
Article

We investigate properties of a fundamental solution and establish the correct solvability of the Cauchy problem for one class of degenerate Kolmogorov-type equations with \( \left\{ {\overrightarrow p, \overrightarrow h } \right\} \)-parabolic part with respect to the main group of variables and with positive vector genus in the case where solutions are infinitely differentiable functions and their initial values may be generalized functions of Gevrey ultradistribution type.

Keywords

Cauchy Problem Fundamental Solution Abstract Function Limit Relation Degenerate Parabolic Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. N. Kolmogoroff, “Zufällige Bewegungen (Zur Theorie der Brownschen Bewegung),” Ann. Math., 35, 116–117 (1934).CrossRefMathSciNetGoogle Scholar
  2. 2.
    S. D. Eidelman, S. D. Ivasyshen, and A. N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Birkhäuser, Basel (2004).zbMATHGoogle Scholar
  3. 3.
    I. G. Petrovskii, “On the Cauchy problem for systems of partial differential equations in a domain of nonanalytic functions,” Byul. Mosk. Univ., Ser. Mat. Mekh., 1, No. 7, 1–72 (1938).Google Scholar
  4. 4.
    S. D. Éidel’man, “On one class of parabolic systems,” Dokl. Akad. Nauk SSSR, 133, No. 1, 40–43 (1960).Google Scholar
  5. 5.
    V. A. Litovchenko, “Cauchy problem for \( \left\{ {\overrightarrow p, \overrightarrow h } \right\} \)-parabolic equations with time-dependent coefficients,” Mat. Zametki, 77, No. 3-4, 364–379 (2005).zbMATHMathSciNetGoogle Scholar
  6. 6.
    I. M. Gel’fand and G. E. Shilov, Some Problems of the Theory of Differential Equations [in Russian], Fizmatgiz, Moscow (1958).Google Scholar
  7. 7.
    I. M. Gel’fand and G. E. Shilov, Spaces of Test and Generalized Functions [in Russian], Fizmatgiz, Moscow (1958).Google Scholar
  8. 8.
    S. D. Ivasyshen and L. M. Androsova, “Localization principle for solutions of certain degenerate parabolic equations,” in: Boundary-Value Problems with Different Degeneracies and Singularities [in Ukrainian], Chernivtsi (1990), pp. 48–61.Google Scholar
  9. 9.
    O. H. Voznyak, “On the unique solvability of the Cauchy problem for one class of degenerate equations in spaces of generalized functions,” Nauk. Visn. Cherniv. Univ., Issue 111, 5–10 (2001).Google Scholar
  10. 10.
    O. H. Voznyak and S. D. Ivasyshen, “Unique solvability and property of localization of solutions of the Cauchy problem for one class of degenerate equations with generalized initial data,” Mat. Met. Fiz.-Mekh. Polya, 44, No. 4, 27–39 (2001).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  • S. D. Ivasyshen
    • 1
  • V. A. Litovchenko
    • 2
  1. 1.“Kyiv Polytechnic Institute” Ukrainian National Technical UniversityKyivUkraine
  2. 2.Chernivtsi National UniversityChernivtsiUkraine

Personalised recommendations