Abstract:
The classical solvability of the singular Cauchy problem for the Euler–Poisson–Darboux equation in a homogeneous, globally symmetric space of rank 1 is studied. Starting out from the mean value theorem for spaces of the indicated type, the Darboux and the Euler–Poisson–Darboux equations are introduced. For the Cauchy problem with specific singularity conditions, analogs of Kirchhoff's formulas are derived, i.e. a representation of the solution in terms of spherical means of the initial data is given. The representations so obtained permitted the establishment of necessary and sufficient conditions for the problems under consideration to satisfy Huygens' principle. In particular, Kirchhoff's formulas for the wave equation have been obtained.
Bibliography: 27 titles.
Citation:
I. A. Kipriyanov, L. A. Ivanov, “The Cauchy problem for the Euler–Poisson–Darboux equation in a symmetric space”, Math. USSR-Sb., 52:1 (1985), 41–51
\Bibitem{KipIva84}
\by I.~A.~Kipriyanov, L.~A.~Ivanov
\paper The Cauchy problem for the Euler--Poisson--Darboux equation in a~symmetric space
\jour Math. USSR-Sb.
\yr 1985
\vol 52
\issue 1
\pages 41--51
\mathnet{http://mi.mathnet.ru/eng/sm2039}
\crossref{https://doi.org/10.1070/SM1985v052n01ABEH002876}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=743056}
\zmath{https://zbmath.org/?q=an:0573.35074}
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https://doi.org/10.1070/SM1985v052n01ABEH002876
https://www.mathnet.ru/eng/sm/v166/i1/p45
This publication is cited in the following 15 articles:
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A. V. Glushak, Trends in Mathematics, Transmutation Operators and Applications, 2020, 379
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Denisov V., “Stabilization of Time Means of a Solution of a Cauchy-Problem for a Singular Hyperbolic Equation”, Differ. Equ., 22:1 (1986), 24–33
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