TY  - JOUR
A1  - Antoniouk, Alexandra Viktorivna
A1  - Kiselev, Oleg M.
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point
JF  - Ukrainian mathematical journal
N2  - The Dirichlet problem for the heat equation in a bounded domain aS, a"e (n+1) is characteristic because there are boundary points at which the boundary touches a characteristic hyperplane t = c, where c is a constant. For the first time, necessary and sufficient conditions on the boundary guaranteeing that the solution is continuous up to the characteristic point were established by Petrovskii (1934) under the assumption that the Dirichlet data are continuous. The appearance of Petrovskii's paper was stimulated by the existing interest to the investigation of general boundary-value problems for parabolic equations in bounded domains. We contribute to the study of this problem by finding a formal solution of the Dirichlet problem for the heat equation in a neighborhood of a cuspidal characteristic boundary point and analyzing its asymptotic behavior.
Y1  - 2015
U6  - https://doi.org/10.1007/s11253-015-1038-8
SN  - 0041-5995
SN  - 1573-9376
VL  - 66
IS  - 10
SP  - 1455
EP  - 1474
PB  - Springer
CY  - New York
ER  - 
TY  - INPR
A1  - Antoniouk, Alexandra Viktorivna
A1  - Kiselev, Oleg
A1  - Stepanenko, Vitaly
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Asymptotic solutions of the Dirichlet problem for the heat equation at a characteristic point
N2  - The Dirichlet problem for the heat equation in a bounded domain is characteristic, for there are boundary points at which the boundary touches a characteristic hyperplane t = c, c being a constant. It was I.G. Petrovskii (1934) who first found necessary and sufficient conditions on the boundary which guarantee that the solution is continuous up to the characteristic point, provided that the Dirichlet data are continuous. This paper initiated standing interest in studying general boundary value problems for parabolic equations in bounded domains. We contribute to the study by constructing a formal solution of the Dirichlet problem for the heat equation in a neighbourhood of a characteristic boundary point and showing its asymptotic character.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)25 
KW  - Heat equation
KW  - the first boundary value problem
KW  - characteristic boundary point
KW  - cusp
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-61987
ER  -