Geodesic
Asymptotics of the fundamental solution of a Petrovskii parabolic equation with constant coefficients
Sbornik. Mathematics, Tome 20 (1973) no. 4, pp. 519-542 
S. G. Gindikin; M. V. Fedoryuk. Asymptotics of the fundamental solution of a Petrovskii parabolic equation with constant coefficients. Sbornik. Mathematics, Tome 20 (1973) no. 4, pp. 519-542. http://geodesic.mathdoc.fr/item/SM_1973_20_4_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $P(\zeta)$, $\zeta\in\mathbf C^n$, be a homogeneous, parabolic polynomial of degree $2m$. Properties of the function $$ \nu(\eta)=\min_{\xi\in\mathbf R^n}\operatorname{Re}P(\xi+i\eta),\qquad\eta\in\mathbf R^n, $$ are investigated. Two-sided estimates are obtained for the fundamental solution $G(t,x)$ of the equation $$ \frac{\partial u}{\partial t}+P\biggl(\frac1i\frac\partial{\partial x}\biggr)u=0, $$ and an asymptotic decomposition is determined for $G(t,x)$ as $|x|^{2m}/t\to+\infty$ under the assumption that $\nu(\eta)\in C^1(\mathbf R^n)$. Bibliography: 14 titles.

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[2] M. V. Fedoryuk, “Asimptotika funktsii Grina pri $t\to+\infty$, $|x|\to\infty$ dlya korrektnykh po Petrovskomu uravnenii s postoyannymi koeffitsientami i klassy korrektnosti resheniya zadachi Koshi”, Matem. sb., 62 (104) (1961), 397–468

[3] S. G. Gindikin, “Trubchatye oblasti i klassy korrektnosti zadachi Koshi”, Uspekhi matem. nauk, XXIII:3 (141) (1968), 177–178 | MR

[4] S. G. Gindikin, “Klassy korrektnosti zadachi Koshi dlya differentsialnykh uravnenii s peremennymi koeffitsientami”, Funkts. analiz, 4:2 (1970), 83–84 | MR | Zbl

[5] S. G. Gindikin, “Svertyvateli v prostranstvakh obobschennykh funktsii s eksponentsialnoi asimptotikoi”, Funkts. analiz, 6:2 (1972), 83–86 | MR

[6] M. A. Evgrafov, M. M. Postnikov, “Asimptotika funktsii Grina parabolicheskikh i ellipticheskikh uravnenii s postoyannymi koeffitsientami”, Matem. sb., 82 (124) (1970), 3–29 | MR | Zbl

[7] V. S. Vladimirov, Metody teorii funktsii mnogikh kompleksnykh peremennykh, izd-vo «Nauka», Moskva, 1966 | MR

[8] Dzh. Milnor, Osobye tochki kompleksnykh giperpoverkhnostei, izd-vo «Mir», Moskva, 1971 | MR

[9] E. A. Gorin, “Ob asimptoticheskikh svoistvakh mnogochlenov i algebraicheskikh funktsii ot neskolkikh peremennykh”, Uspekhi matem. nauk, XVI:1 (97) (1961), 91–118 | MR

[10] Zh. De Ram, Differentsiruemye mnogoobraziya, IL, Moskva, 1956

[11] I. R. Shafarevich, Osnovy algebraicheskoi geometrii, izd-vo «Nauka», Moskva, 1972 | MR

[12] E. B. Vinberg, A. A. Onischik, Seminar po algebraicheskim gruppam i gruppam Li, izd-vo MGU, Moskva, 1969

[13] S. G. Gindikin, M. V. Fedoryuk, “Asimptotika fundamentalnogo resheniya parabolicheskogo uravneniya s postoyannymi koeffitsientami”, Uspekhi matem. nauk, XXVIII:1 (169) (1973), 235–236 | MR

[14] V. F. Demyanov, V. N. Malozemov, Vvedenie v minimaks, izd-vo «Nauka», Moskva, 1972 | MR