Literature Cited
E. Holmgren, “Sur les solutions quasianalitiques de l'equations de la chaleur,” Arkiv Mat.,18, 64–95 (1924).
A. N. Tikhonov, “Uniqueness theorems for heat equations,” Mat. Sb.,42, No. 2, 199–216 (1935).
S. Täcklind, “Sur les classes quasianalitiques des solutions des l'equations aux derivées partielles du type parabolique,” Nord. Acta. Regial. Sociatis schientiarum Uppsaliensis, Ser. 4,10, No. 3, 3–55 (1936).
I. G. Petrovskii, “Some problems in the theory of equations in partial derivatives,” Usp. Mat. Nauk,1, Nos. 3–4, 44–70 (1946).
A. M. Il'in, A. S. Kalashnikov, and O. A. Oleinik, “Linear second-order parabolic equations,” Usp. Mat. Nauk,17, No. 3, 3–146 (1962).
O. A. Oleinik and E. V. Radkevich, “The method of introducing a parameter for studying evolutionary equations,” Usp. Mat. Nauk,33, No. 5, 7–76 (1978).
M. Krzyzanski, “Sur les solutions des equations du type parabolique determinées dans une region illimitee, Bull. Am. Math. Soc.,47, 911–915 (1941).
R. M. Hayne, “Uniqueness in the Cauchy problem for parabolic equations,” Trans. Am. Math. Soc.,241, 373–399 (1978).
I. M. Gel'fand and G. E. Shilov, “Fourier transforms of quickly growing functions and problems of the uniqueness of solutions of Cauchy's problem,” Usp. Mat. Nauk,8, No. 6, 3–54 (1953).
N. I. Chaus, “Uniqueness classes of a solution of Cauchy's problem, and the representation of positivedefinite kernels,” in: Proceedings of the Seminar on Functional Analysis [in Russian], Vol. 1, Izd. Inst. Mat. Akad. Nauk Ukr. SSR, Kiev (1968), pp. 176–273.
G. N. Smirnova, “Cauchy's problem for parabolic equations degenerate at infinity,” Mat. Sb.,70, No. 4, 591–604 (1966).
I. M. Sonii, “Uniqueness classes for degenerate parabolic equations,” Mat. Sb.,85, No. 5, 459–473 (1971).
L. I. Kamynin and B. N. Khimchenko, “The uniqueness of the solution of Cauchy's problem for a secondorder parabolic equation with nonnegative characteristic form,” Dokl. Akad. Nauk SSSR,248, No. 2, 290–293 (1979).
L. I. Kamynin and B. N. Khimchenko, “Tikhonov-Täcklind uniqueness classes for degenerate secondorder parabolic equations,” Dokl. Akad. Nauk SSSR,252, No. 4, 784–788 (1980).
L. I. Kamynin and B. N. Khimchenko, “The local behavior of solutions of a second-order parabolic equation near to the lower cover of the parabolic boundary,” Sib. Mat. Zh.,20, No. 1, 69–94 (1979).
L. I. Kamynin and B. N. Khimchenko, “A priori estimates of solutions of second-order parabolic equations near to the lower cover of the parabolic boundary,” Sib. Mat. Zh.,22, No. 4, 94–113 (1981).
A. S. Kalashnikov, “Increasing solutions of linear second-order equations with nonstationary characteristic form,” Mat. Zametki,3, No. 2, 171–178 (1968).
Ya. I. Zhitomirskii, “Cauchy's problem for a second-order parabolic equation with variable coefficients,” Dokl. Akad. Nauk SSSR,116, No. 6, 913–916 (1957).
Ya. I. Zhitomorskii, “Exact uniqueness classes for solutions of Cauchy's problem for second-order equations,” Dokl. Akad. Nauk SSSR,171, No. 1, 29–32 (1967).
Ya. I. Zhitomorskii, “Uniqueness classes for solutions of Cauchy's problem for linear equations with quickly growing coefficients,” Izv. Akad. Nauk SSSR, Ser. Mat.,31, No. 5, 1159–1178 (1967).
Ya. I. Zhitomirskii, “Uniqueness classes for solutions of Cauchy's problem for linear equations with increasing coefficients,” Izv. Akad. Nauk SSSR, Ser. Mat.,31, No. 4, 762–782 (1967).
A. S. Kalashnikov, “Linear degenerate parabolic equations of arbitrary order with finite domain of dependence,” Mat. Zametki,6, No. 3, 289–294 (1969).
V. Filler, “The theory of stochastic processes,” Usp. Mat. Nauk,5, 57–96 (1938).
S. D. Eidelman, Parabolic Systems, Elsevier (1969).
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M. V. Lomonosov State University. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 22, No. 5, pp. 78–109, September–October, 1981.
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Kamynin, L.I., Khimchenko, B.N. Tikhonov-Petrovskii problem for second-order parabolic equations. Sib Math J 22, 709–734 (1981). https://doi.org/10.1007/BF00968068
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DOI: https://doi.org/10.1007/BF00968068