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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. N. Kolmogoroff, “Zufällige Bewegungen (Zur Theorie der Brownschen Bewegung),” Ann. Math., 35, 116–117 (1934).
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).
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).
S. D. Éidel’man, “On one class of parabolic systems,” Dokl. Akad. Nauk SSSR, 133, No. 1, 40–43 (1960).
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).
I. M. Gel’fand and G. E. Shilov, Some Problems of the Theory of Differential Equations [in Russian], Fizmatgiz, Moscow (1958).
I. M. Gel’fand and G. E. Shilov, Spaces of Test and Generalized Functions [in Russian], Fizmatgiz, Moscow (1958).
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.
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).
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).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 8, pp. 1066–1087, August, 2009.
Rights and permissions
About this article
Cite this article
Ivasyshen, S.D., Litovchenko, V.A. Cauchy problem for one class of degenerate parabolic equations of Kolmogorov type with positive genus. Ukr Math J 61, 1264–1288 (2009). https://doi.org/10.1007/s11253-010-0275-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-010-0275-0