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Uspekhi Mat. Nauk, 1972, Volume 27, Issue 4(166), Pages 65–143 (Mi umn5084)  

This article is cited in 10 scientific papers (total in 10 papers)

The cauchy problem and other related problems for convolution equations

L. R. Volevich, S. G. Gindikin

Abstract: Spaces of generalized functions with exponential asymptotic behaviour are considered. Convolutors in these spaces are completely described. It is shown that a convolution equation is uniquely soluble if and only if there exists a fundamental solution that is a convolutor. The explicit description of convolutors renders this condition effective. In particular, Petrovskii's correctness condition is obtained in the case of differential equations. A calculus of pseudodifferential operators with inhomogeneous symbols of constant strength is constructed; the solubility of the Cauchy problem can be proved by means of this calculus for a certain class of differential equations with variable coefficients.

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English version:
Russian Mathematical Surveys, 1972, 27:4, 71–160

Bibliographic databases:

UDC: 517.9
MSC: 45E10, 42A85, 42A38, 35Sxx
Received: 02.01.1972

Citation: L. R. Volevich, S. G. Gindikin, “The cauchy problem and other related problems for convolution equations”, Uspekhi Mat. Nauk, 27:4(166) (1972), 65–143; Russian Math. Surveys, 27:4 (1972), 71–160

Citation in format AMSBIB
\by L.~R.~Volevich, S.~G.~Gindikin
\paper The cauchy problem and other related problems for convolution equations
\jour Uspekhi Mat. Nauk
\yr 1972
\vol 27
\issue 4(166)
\pages 65--143
\jour Russian Math. Surveys
\yr 1972
\vol 27
\issue 4
\pages 71--160

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    This publication is cited in the following articles:
    1. L. R. Volevich, “Energeticheskie otsenki i zadacha Koshi dlya differentsialnykh operatorov s peremennymi koeffitsientami v prostranstvakh funktsii s eksponentsialnoi asimptotikoi”, UMN, 29:1(175) (1974), 167–168  mathnet  mathscinet  zmath
    2. V. Ya. Ivrii, V. M. Petkov, “Necessary conditions for the Cauchy problem for non-strictly hyperbolic equations to be well-posed”, Russian Math. Surveys, 29:5 (1974), 1–70  mathnet  crossref  mathscinet  zmath
    3. A. L. Pavlov, “On general boundary value problems for differential equations with constant coefficients in a half-space”, Math. USSR-Sb., 32:3 (1977), 313–334  mathnet  crossref  mathscinet  zmath  isi
    4. O. A. Oleinik, E. V. Radkevich, “The method of introducing a parameter in the study of evolutionary equations”, Russian Math. Surveys, 33:5 (1978), 7–84  mathnet  crossref  mathscinet  zmath
    5. L. R. Volevich, S. G. Gindikin, “The method of energy estimates in mixed problems”, Russian Math. Surveys, 35:5 (1980), 57–137  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. A. L. Pavlov, “The Cauchy problem for Sobolev–Gal'pern type equations in spaces of functions of power growth”, Russian Acad. Sci. Sb. Math., 80:2 (1995), 255–269  mathnet  crossref  mathscinet  zmath  isi
    7. A. L. Pavlov, “Solvability of boundary value problems in a half-space for differential equations with constant coefficients in the class of tempered distributions”, Siberian Math. J., 54:4 (2013), 697–712  mathnet  crossref  mathscinet  isi
    8. A. L. Pavlov, “On regularization of a certain class of distributions”, Math. Nachr, 2015, n/a  crossref
    9. A. L. Pavlov, “On the division problem for a tempered distribution that depends holomorphically on a parameter”, Siberian Math. J., 56:5 (2015), 901–911  mathnet  crossref  crossref  isi  elib
    10. A. L. Pavlov, “Zadacha Koshi dlya odnogo uravneniya sobolevskogo tipa v klasse obobschennykh funktsii medlennogo rosta”, Matem. tr., 21:1 (2018), 125–154  mathnet  crossref  elib
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