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 Izv. RAN. Ser. Mat., 1997, Volume 61, Issue 3, Pages 91–132 (Mi izv127)

Representative systems of exponentials and the Cauchy problem for partial differential equations with constant coefficients

Yu. F. Korobeinik

Rostov State University

Abstract: We consider the Cauchy problem with respect to $z_2$ for a homogeneous linear partial differential equation with constant coefficients in two independent variables $z_1,z_2 \in \mathbb C$. We show that the relative smoothness with respect to $z_1$ and $z_2$ of analytic and ultradifferentiable solutions of the Cauchy problem depends essentially on the value of $\rho_2$ and, as a rule, is completely determined by it. We also obtain rather general uniqueness theorems and find conditions which guarantee that the particular solution constructed depends both continuously and linearly on the initial functions.

DOI: https://doi.org/10.4213/im127

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English version:
Izvestiya: Mathematics, 1997, 61:3, 553–592

Bibliographic databases:

MSC: 35E15, 46E10

Citation: Yu. F. Korobeinik, “Representative systems of exponentials and the Cauchy problem for partial differential equations with constant coefficients”, Izv. RAN. Ser. Mat., 61:3 (1997), 91–132; Izv. Math., 61:3 (1997), 553–592

Citation in format AMSBIB
\Bibitem{Kor97} \by Yu.~F.~Korobeinik \paper Representative systems of exponentials and the Cauchy problem for partial differential equations with constant coefficients \jour Izv. RAN. Ser. Mat. \yr 1997 \vol 61 \issue 3 \pages 91--132 \mathnet{http://mi.mathnet.ru/izv127} \crossref{https://doi.org/10.4213/im127} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1478561} \zmath{https://zbmath.org/?q=an:0903.35013} \transl \jour Izv. Math. \yr 1997 \vol 61 \issue 3 \pages 553--592 \crossref{https://doi.org/10.1070/IM1997v061n03ABEH000127} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1997YH77500004} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-21944437463} 

• http://mi.mathnet.ru/eng/izv127
• https://doi.org/10.4213/im127
• http://mi.mathnet.ru/eng/izv/v61/i3/p91

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Erratum

This publication is cited in the following articles:
1. Yu. F. Korobeinik, “Letter to the Editors”, Izv. Math., 62:3 (1998), 649–649
2. Korobeinik Y.F., “The Fourier method in the Cauchy problem and absolutely representing systems of exponentials. I”, Differential Equations, 35:12 (1999), 1693–1701
3. Korobeinik Y.F., “Absolutely representing systems of exponentials with imaginary exponents in spaces of infinitely differentiable functions”, Doklady Mathematics, 61:3 (2000), 324–327
4. Korobeinik Y.F., “On absolutely representing systems in spaces of infinitely differentiable functions”, Studia Mathematica, 139:2 (2000), 175–188
5. Korobeinik Y.F., “The Fourier method in the Cauchy problem and absolutely representing systems of exponentials: III”, Differential Equations, 36:3 (2000), 433–440
6. Korobeinik Y.F., “The Fourier method in the Cauchy problem and absolutely representing systems of exponentials: II”, Differential Equations, 36:2 (2000), 285–290
7. Alexander A. Znamenskiy, “A refinement of Kovalevskaya's theorem on analytic solvability of the Cauchy problem”, Zhurn. SFU. Ser. Matem. i fiz., 10:4 (2017), 531–536
8. S. N. Melikhov, “Koeffitsienty ryadov eksponent dlya analiticheskikh funktsii i operator Pomme”, Kompleksnyi analiz. Tselye funktsii i ikh primeneniya, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 161, VINITI RAN, M., 2019, 65–103
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