Email this article 
HE CAUCHY PROBLEM FOR PARABOLIC SYSTEM WITH VARIABLE COEFFICIENTS IN ANISOTROPIC ZYGMUND SPACES
- Authors: Egorova A.Y.1, Konenkov A.N1
-
Affiliations:
- Ryazan State University named after S.A. Esenin
- Issue: Vol 60, No 10 (2024)
- Pages: 1325-1333
- Section: PARTIAL DERIVATIVE EQUATIONS
- URL: https://ter-arkhiv.ru/0374-0641/article/view/649599
- DOI: https://doi.org/10.31857/S0374064124100039
- EDN: https://elibrary.ru/JTSXEW
- ID: 649599
Cite item
Open Access
Access granted
Subscription Access
Abstract
The Cauchy problem for a second-order parabolic system with coefficients and the right hand side which belong to the Zygmund anisotropic space is considered. A smoothness scale of the Cauchy problem solutions in anisotropic Zygmund spaces is obtained. A priori estimates of solutions for uniformly elliptic systems in isotropic Zygmund spaces are derived.
About the authors
A. Yu Egorova
Ryazan State University named after S.A. Esenin
Email: an_batseva@mail.ru
Russia
A. N Konenkov
Ryazan State University named after S.A. Esenin
Email: a.konenkov@365.rsu.edu.ru
Russia
References
- Ladyzhenskaya, O.A., Solonnikov, V.A., and Ural’tseva, N.N., Linear and Quasi-Linear Equations of Parabolic Type, Providence: AMS, 1968.
- Gilbarg, D. and Trudinger, N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1977.
- Konenkov, A.N., The Cauchy problem for the heat equation in Zygmund spaces, Differ. Equat., 2005, vol. 41, no. 6, pp. 860–872.
- Konenkov, A.N., The Cauchy problem for parabolic equations in Zygmund spaces, Differ. Equat., 2006, vol. 42, no. 6, pp. 867–873.
- Solonnikov, V.A., On boundary value problems for linear parabolic systems of differential equations of general form, Trudy Mat. Inst. Steklov., 1965, vol. 83, pp. 3–163.
- Cherepova, M.F., On the smoothness of the solution of Cauchy’s problem for the parabolic system, Bulletin MPEI, 2009, no. 6, pp. 38–44.
- Baderko, E.A. and Cherepova, M.F., Uniqueness theorem for parabolic Cauchy problem, Appl. Anal., 2016, vol. 95, no. 7, pp. 1570–1580.
- Baderko, E.A. and Cherepova, M.F., Uniqueness of the solution of a the Cauchy problem for parabolic systems, Differ. Equat., 2019, vol. 55. no. 6, pp. 806–814.
- Baderko, E.A. and Fedorov, K.D., On the smoothness of the Poisson potential for second-order parabolic systems on the plane, Differ. Equat., 2023, vol. 59, no. 12, pp. 1613–1626.
- Egorova, A.Yu., The Cauchy problem for parabolic system of equations in anisotropic Zydmund spaces, Bulletin of BSU. Mathematics, computer science, 2023, no. 3, pp. 14–22.
- Egorova, A.Yu., The Cauchy problem for inhomogeneous parabolic systems in anisotropic Zydmund spaces, Bulletin of the South Ural State Univ. Series: Mathematics. Mechanics. Physics, 2024, vol. 16, no. 1, pp. 5–12.
- Eidel’man, S.D., Parabolic Systems, Elsevier, 1969.
- Konenkov, A.N., On the relationship between fundamental solutions of elliptic and parabolic equations, Differ. Equat., 2002, vol. 38, no. 2, pp. 263–273.
- Triebel, H., Theory of Function Spaces. Basel: Birkh¨auser, 1983.
- Besov, O.V., Ilyin, V.P., and Nikol’sky, S.M. Integral Representation of Functions and Embedding Theorems, Washington DC: V.H. Wilson and Sons, 1978.
Supplementary files
