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

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

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
\Bibitem{VolGin72} \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 \mathnet{http://mi.mathnet.ru/umn5084} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=393908} \zmath{https://zbmath.org/?q=an:0244.35016} \transl \jour Russian Math. Surveys \yr 1972 \vol 27 \issue 4 \pages 71--160 \crossref{https://doi.org/10.1070/RM1972v027n04ABEH003368} 

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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
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
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
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
5. L. R. Volevich, S. G. Gindikin, “The method of energy estimates in mixed problems”, Russian Math. Surveys, 35:5 (1980), 57–137
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
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
8. A. L. Pavlov, “On regularization of a certain class of distributions”, Math. Nachr, 2015, n/a
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
10. A. L. Pavlov, “Zadacha Koshi dlya odnogo uravneniya sobolevskogo tipa v klasse obobschennykh funktsii medlennogo rosta”, Matem. tr., 21:1 (2018), 125–154
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