In the case where initial data are generalized functions of the Gevrey-distribution type for which the classical notion of equality of two functions on a set is well defined, we establish the principle of local strengthening of the convergence of a solution of the Cauchy problem to its limit value as t → +0 for one class of degenerate parabolic equations of the Kolmogorov type with \( \overrightarrow {2b} \)-parabolic part whose coefficients are continuous functions that depend only on t.
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–Boston–Berlin (2004).
O. G. Voznyak and S. D. Ivasyshen, “Unique solvability and 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).
I. M. Gel’fand and G. E. Shilov, Spaces of Fundamental and Generalized Functions [in Russian], Fizmatgiz, Moscow (1958); English translation: Academic Press, New York–London (1968).
S. D. Ivasyshen and V. A. Litovchenko, “Cauchy problem for one class of degenerate parabolic equations of Kolmogorov type with positive genus,” Ukr. Mat. Zh., 61, No. 8, 1066–1087 (2009); English translation: Ukr. Math. J., 61, No. 8, 1264–1288 (2009).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 11, pp. 1473–1489, November, 2010.
Rights and permissions
About this article
Cite this article
Litovchenko, V.A., Strybko, O.V. Principle of localization of solutions of the Cauchy problem for one class of degenerate parabolic equations of Kolmogorov type. Ukr Math J 62, 1707–1728 (2011). https://doi.org/10.1007/s11253-011-0462-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-011-0462-7