Fundamental solution of the Cauсhy problem for the dissipative ultraparabolic Kolmogorov equation with coefficients independent on variable degeneration
DOI:
https://doi.org/10.17721/1812-5409.2025/2.20Keywords:
increasing coefficients, fundamental solution to the Cauchy problem, the Levy method, ultraparabolic Kolmogorov equationAbstract
For the ultraparabolic Kolmogorov equation with coefficients that do not depend on the degeneracy variables and are unbounded in the spatial variable, a fundamental solution of the Cauchy problem is constructed. The growth of the coefficients of the equation depends on an increasing function, which is called the characteristic of dissipation. The coefficients of the equation satisfy the special Hölder condition, which contains the characteristic of dissipation. Differentiability of the coefficients is not required; therefore, an additional condition is imposed on the characteristic of dissipation. Ultraparabolic Kolmogorov equations with increasing coefficients arise in problems of the theory of random processes, statistical radio engineering.
The fundamental solution of the Cauchy problem is constructed by the Levy method. In this case, the parametrics are taken as the fundamental solution of the Cauchy problem of the equation under consideration with coefficients depending on the time and parametric spatial variables. This complicates the study, since in the case of bounded coefficients, the parametrics are taken as the fundamental solution of the Cauchy problem of the equation with frozen coefficients only of the group of higher derivatives.
The obtained estimates of the derivatives of the fundamental solution of the Cauchy problem can be used in studying the issues of solvability of the Cauchy problem and the integral representation of solutions in special weight spaces, as well as for constructing a classical fundamental solution of the Cauchy problem for a degenerate equation of the Kolmogorov type with coefficients increasing as |x| -> ∞, which also depend on the degeneracy variables, and systems of such equations.
Pages of the article in the issue: 130 - 137
Language of the article: Ukrainian
References
Eidelman, S. D., Ivasyshen, S. D., & Kochubei, A. N. (2004). Analytic methods in the theory of differential and pseudo-differential equations of parabolic type. Birkhäuser. https://doi.org/10.1007/978-3-0348-7844-9
Di Francesco, M., & Pascucci, A. (2005). On a class of degenerate parabolic equations of Kolmogorov type. Applied Mathematics Research eXpress, 3, 77–116. https://doi.org/10.1155/AMRX.2005.77
Ivasyshen, S. D., & Medyns'kyi, I. P. (2014). The classical fundamental solution of a degenerate Kolmogorov equation with coefficients independent on variables of degeneration. Bukovinian Mathematical Journal, 2(2–3), 94–106 [in Ukrainian].
Ivasyshen, S. D., & Medyns'kyi, I. P. (2018). On the classical fundamental solutions of the Cauchy problem for ultraparabolic Kolmogorov-type equations with two groups of spatial variables. Journal of Mathematical Sciences, 231(4). 507–526. https://doi.org//10.1007/s10958-018-3830-0
Ivasyshen, S. D., & Pasichnyk, H. S. (2014). Fundamental solution of the Cauchy problem for parabolic equation with growing lowest coefficients. Proceedings of Institute of Mathematics NAS of Ukraine, 11(2), 126–153 [in Ukrainian].
Pasichnyk, H. S. (2024). Solutions of some integral equations of the second kind. Bukovinian Mathematical Journal, 12(1), 84–93 [in Ukrainian]. https://doi.org/10.31861/bmj2024.01.08
Protsach, N. P., & Ptashnyk, B. Yo. (2017). Nonlinear ultraparabolic equations and variational inequalities. Naukova dumka [in Ukrainian].
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Halyna Pasichnyk
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
