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Propagation des singularités pour des équations hyperboliques à caractéristiques de multiplicité au plus double et singularités Masloviennes. II. (Propagation of singularities for hyperbolic equations with characteristic of multiplicity greater than two and Maslov singularities. II). (French) Zbl 0592.58053
[For Part I see the first author, Am. J. Math. 104, 227-285 (1982; Zbl 0506.35067).]
The authors construct explicit solutions to hyperbolic equations with at most double characteristics; the equations are noneffectively hyperbolic according to the classification of V. Ya. Ivrij and V. Petkov [Usp. Mat. Nauk 29, No.5(179), 3-70 (1974; Zbl 0312.35049)] and L. Hörmander [J. Anal. Math. 32, 118-196 (1977; Zbl 0367.35054)]. The double characteristics N form a noninvolutive \(C^{\infty}\) manifold. They study the case \(p=D^ 2_ 0-A(x,D_ x)\), where A is a classical pseudodifferential operator with nonnegative principal symbol which vanishes exactly of order two on a symplectic submanifold of \(T^*X\setminus O\). They suppose that a positivity condition on the subprincipal symbol and on the fundamental matrix is satisfied in order to have a well-posed Cauchy problem. They used first author’s method from part I. The explicit solution is used to study the propagation of singularities.
Reviewer: R.Vaillancourt

58J47 Propagation of singularities; initial value problems on manifolds
Full Text: DOI
[1] S. Alinhac,Paramétrixe pour un système hyperbolique à multiplicité variable, Commun. Partial Differ. Equ.2 (3) (1977), 251–296. · Zbl 0355.35057
[2] S. Alinhac,Solution explicite du probleme de Cauchy pour des óperateurs effectivement hyperboliques, Duke Math. J.45 (1978), 225–258. · Zbl 0398.35061
[3] R. Beals,Propagation of singularities of solutions of D 1 2 -, Colloque de St Jean de Monts, Juin 1980. · Zbl 0454.58018
[4] L. Boutet de Monvel,Hypoelliptic operators with double characteristics and related pseudo-differential operators, Commun. Pure Appl. Math.27 (1974), 585–639. · Zbl 0294.35020
[5] J. J. Duistermaat and L. Hörmander,Fourier integral operator II, Acta Math.128 (1972), 183–269. · Zbl 0232.47055
[6] G. Eskin,Parametrix and propagation of singularities for the interior mixed hyperbolic problem, J. Analyse Math.32 (1977), 17–62. · Zbl 0375.35037
[7] L. Hörmander,Fourier integral operator, Acta Math.127 (1971), 79–183. · Zbl 0212.46601
[8] L. Hörmander,The Cauchy problem for differential equations with double characteristics, J. Analyse Math.32 (1977), 118–196. · Zbl 0367.35054
[9] V. J. Ivri,Wave front of solutions of some hyperbolic equations and conical refraction, Sov. Math. Dokl.226 (1976).
[10] V. J. Ivri,Wave front of some hyperbolic equations, Sov. Math. Dokl.226, No. 5 and229, No. 2 (1976).
[11] V. J. Ivri, Trudi Moskov.39 (1979), 49–81 et 83–112.
[12] V. J. Ivri and V. Petkov,Necessary conditions for correctness of the Cauchy problem, Uspehi Mat. Nauk no 5 (1974).
[13] B. Lascar,Propagations des singularités pour des équations hyperboliques à caractéristique de multiplicité au plus double et singularités Masloviennes I, à paraître dans Am. J. Math.
[14] R. Lascar,Parametrixes microlocales du problème aux limites pour une classe d’équations pseudo-différentielles à caractéristiques de multiplicité variables, C. R. Acad. Sci. Paris287 (1978); Thèse Doctorat d’Etat, Paris 7, 1979.
[15] R. Lascar,Relation entre l’unicité du problème de Cauchy et le problème des singularités d’équations pseudo-différentielles à caractéristiques involutives de multiplicité variables, à paraître; Thèse de Doctorat d’Etat, Paris 7, 1979.
[16] R. Melrose and J. Ulhmann, preprint, 1979.
[17] R. Melrose,Hypoelliptic operators with characteristics variety of codimension two and the wave equation, Séminaire Goulaouic-Schwartz, Ecole Polytechnique, Paris, Janvier 1980. · Zbl 0468.35032
[18] A. Menikoff and J. Sjöstrand,On the eigen values of a class of hypoelliptic operators, Math. Ann.235 (1978), 55–85. · Zbl 0375.35014
[19] J. Sjöstrand,Operators of principal type with interior boundary conditions, Acta Math.130 (1973), 1–51. · Zbl 0253.35076
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