@article{AntonioukKiselevTarkhanov2015,
  author    = {Antoniouk, Alexandra Viktorivna and Kiselev, Oleg M. and Tarkhanov, Nikolai Nikolaevich},
  title     = {Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point},
  series = {Ukrainian mathematical journal},
  volume    = {66},
  journal   = {Ukrainian mathematical journal},
  number    = {10},
  publisher = {Springer},
  address   = {New York},
  issn      = {0041-5995},
  doi       = {10.1007/s11253-015-1038-8},
  pages     = {1455 -- 1474},
  year      = {2015},
  abstract  = {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.},
  language  = {en}
}
@unpublished{AntonioukKiselevStepanenkoetal.2012,
  author    = {Antoniouk, Alexandra Viktorivna and Kiselev, Oleg and Stepanenko, Vitaly and Tarkhanov, Nikolai Nikolaevich},
  title     = {Asymptotic solutions of the Dirichlet problem for the heat equation at a characteristic point},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-61987},
  year      = {2012},
  abstract  = {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.},
  language  = {en}
}