We consider the second initial-boundary value problem for a Petrovskii parabolic second order system with variable coefficients in a semibounded plane domain with nonsmooth lateral boundary. Using the boundary integral equation method, we show that the solution belongs to the class \({C}_{x,t}^{\mathrm{2,1}}\) (\(\overline{\Omega }\)) and construct the solution in the form of a special parabolic potential.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. A. Solonnikov, “On boundary value problems for linear parabolic systems of differential equations of general form,” Proc. Steklov Inst. Math. 83 (1965).
E. A. Baderko, “Schauder estimates for oblique derivative problems,” C. R. Acad. Sci., Paris, Sér. I. Math. 326, No. 12, 1377–1380 (1998).
V. A. Tveritinov, “On the second boundary value problem for a parabolic system with one space variable” [in Russian], Differ. Uravn. 26, No.1, 174–175 (1990).
Kh. Semaan, “Estimates of the solution of the second boundary value problem for parabolic systems in the Hölder space \({\underset{o}{\mathrm{C}}}_{\uplambda }^{2+\alpha ,\left(2+\alpha \right)/2}\left(\overline{\Omega }\right)\),” Differ. Equ. 37, No. 1, 146–148 (2001).
E. A. Baderko and A. A. Stasenko, “Smooth solution of the second initial-boundary value problem for a model parabolic system in a semibounded nonsmooth domain on the plane,” Comput. Math. Math. Phys. 62, No. 3, 382–392 (2022).
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).
E. A. Baderko and M. F. Cherepova, “The first boundary value problem for parabolic systems in plane domains with nonsmooth lateral boundaries,” Dokl. Math. 90, No. 2, 573–575 (2014).
E. A. Baderko and M. F. Cherepova, “Simple layer potential and the first boundary value problem for a parabolic system on the plane,” Differ. Equ. 52, No. 2, 197–209 (2016).
Kh. Semaan, “Solvability in the Hölder space \({\underset{o}{\mathrm{C}}}_{\uplambda }^{1+\alpha ,\left(1+\alpha \right)/2}\left(\overline{\Omega }\right)\) of the Neumann boundary value problem for parabolic systems in a domain with a nonsmooth lateral boundary,” Mosc. Univ. Math. Bul. 55, No. 2, 29–31 (2000).
L. I. Kamynin, “Gevrey’s theory of parabolic potentials. V,” Differ. Equations 8, 373–384 (1974).
E. A. Baderko and M. F. Cherepova, “Uniqueness of solutions of the first and second initialboundary value problems for parabolic systems in bounded domains on the plane,” Differ. Equ. 57, No. 8, 1010–1019 (2021).
E. A. Baderko and S. I. Sakharov, “Uniqueness of solutions of initial-boundary value problems for parabolic systems with Dini-continuous coefficients in domains on the plane,” Dokl. Math. 105, No. 2, 71–74 (2022).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Problemy Matematicheskogo Analiza 120, 2023, pp. 5-18.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Baderko, E.A., Cherepova, M.F. Smooth Solution to the Second Initial-Boundary Value Problem for a Parabolic System in a Domain with Nonsmooth Lateral Boundary. J Math Sci 269, 1–17 (2023). https://doi.org/10.1007/s10958-023-06251-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-023-06251-1