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The Cauchy problem for differential equations with double characteristics. (English) Zbl 0367.35054


MSC:

35S10 Initial value problems for PDEs with pseudodifferential operators
35L30 Initial value problems for higher-order hyperbolic equations
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References:

[1] Duistermaat, J. J.; Hörmander, L., Fourier integral operators II, Acta Math., 128, 183-269 (1972) · Zbl 0232.47055
[2] Hörmander, L., Linear Partial Differential Operators (1963), Berlin-Göttingen-Heidelberg: Springer-Verlag, Berlin-Göttingen-Heidelberg · Zbl 0108.09301
[3] Hörmander, L., Pseudo-differential operators and non-elliptic boundary problems, Ann. of Math., 83, 129-209 (1966) · Zbl 0132.07402
[4] Hörmander, L., Fourier integral operators. I., Acta Math., 127, 79-183 (1971) · Zbl 0212.46601
[5] Hörmander, L., A class of hypoelliptic pseudodifferential operators with double characteristics, Math. Ann., 217, 165-188 (1975) · Zbl 0306.35032
[6] Ivrii, V. Ia., Sufficient conditions for regular and completely regular hyperbolicity, Trudy Moskov. Mat. Obšč., 33, 1-65 (1975) · Zbl 0376.35038
[7] Ivrii, V. Ia., Energy integrals for non-strictly hyperbolic operators, Uspehi Mat. Nauk, 30, 6, 169-170 (1975) · Zbl 0315.35061
[8] Ivrii, V. Ia.; Petkov, V. M., Necessary conditions for the correctness of the Cauchy problem for non-strictly hyperbolic equations, Uspehi Mat. Nauk, 29, 5, 3-70 (1974) · Zbl 0312.35049
[9] H. Kumano-go,Factorizations and fundamental solutions for differential operators of elliptic-hyperbolic type, Mimeographed manuscript.
[10] Lax, A., On Cauchy’s problem for partial differential equations with multiple characteristics, comm. Pure Appl. Math., 9, 135-169 (1956) · Zbl 0073.31701
[11] Lax, P. D., Asymptotic solutions of oscillatory initial value problems, Duke Math. J., 24, 627-646 (1957) · Zbl 0083.31801
[12] Lax, P. D.; Nirenberg, L., On stability for difference schemes; a sharp form of Gårding’s inequality, Comm. Pure Appl. Math., 19, 473-492 (1966) · Zbl 0185.22801
[13] Levi, E. E., Caratteristiche multiple e problema di Cauchy, Ann. di Matem. Ser. 3, 16, 161-201 (1909) · JFM 40.0415.02
[14] Melin, A., Lower bounds for pseudo-differential operators, Ark. Mat., 9, 117-140 (1971) · Zbl 0211.17102
[15] Mizohata, S., Some remarks on the Cauchy problem, J. Math. Kyoto Univ., 1, 109-127 (1961) · Zbl 0104.31903
[16] Nirenberg, L.; Treves, F., On local solvability of linear partial differential equations. Part II, Sufficient conditions, Comm. Pure Appl. Math., 23, 459-509 (1970) · Zbl 0208.35902
[17] Olejnik, O. A., On the Cauchy problem for weakly hyperbolic equations, Comm. Pure Appl. Math., 23, 569-586 (1970)
[18] Petkov, V. M., The Cauchy problem for a class of non-strictly hyperbolic equations with double characteristics, Serdica, Bulg. Mat. Publ., 1, 372-380 (1975) · Zbl 0442.35046
[19] Petrowsky, I. G., Über das Cauchysche Problem für ein System linearer partieller Differentialgleichungen, Mat. Sb., 2, 44, 815-868 (1937) · Zbl 0018.40503
[20] Schwid, N., The asymptotic forms of the Hermite and Weber functions, Trans. Amer. Math. Soc., 37, 339-362 (1935) · Zbl 0011.21402
[21] Segal, I., Transforms for operators and symplectic automorphisms over a locally compact abelian group, Math. Scand., 13, 31-43 (1963) · Zbl 0208.39002
[22] Sjöstrand, J., Parametrices for pseudodifferential operators with multiple characteristics, Ark. Mat., 12, 85-130 (1974) · Zbl 0317.35076
[23] Svensson, L., Necessary and sufficient conditions for the hyperbolicity of polynomials with hyperbolic principal part, Ark. Mat., 8, 145-162 (1969) · Zbl 0203.40903
[24] Weil, A., Sur certains groupes d’opérateurs unitaires, Acta Math., 111, 143-211 (1964) · Zbl 0203.03305
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