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The effectively hyperbolic Cauchy problem

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The Hyperbolic Cauchy Problem

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1505))

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References

  1. M.F. Atiyah, R. Bott, L. Gårding: Lacunas for hyperbolic differential operators with constant coefficients, I. Acta Math. 124 (1970), 109–189.

    Article  MathSciNet  MATH  Google Scholar 

  2. C. Briot, J.C. Bouquet: Recherches sur les propriétés des fonctions définies par des équations différentielles. J. Ec. Imp. Polytech. 36 (1856), 133–198.

    Google Scholar 

  3. J.J. Duistermaat: Fourier Integral Operators. Lecture Notes, Courant Institute of Mathematical Sciences, New York, 1973.

    MATH  Google Scholar 

  4. F.R. de Hoog, R. Weiss: Difference methods for boundary value problems with a singularity of the first kind. SIAM J. Numer. Anal. 13 (1976), 775–813.

    Article  MathSciNet  MATH  Google Scholar 

  5. F.R. de Hoog, R. Weiss: On the boundary value problem for systems of ordinary differential equations with a singularity of the second kind. SIAM J. Math. Anal. 11 (1980), 41–60.

    Article  MathSciNet  MATH  Google Scholar 

  6. L. Hörmander: On the singularities of solutions of partial differential equations. Comm. Pure Appl. Math. 23 (1970), 329–358.

    Article  MathSciNet  MATH  Google Scholar 

  7. L. Hörmander: Fourier integral operators, I. Acta Math. 127 (1971), 79–183.

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Hörmander: On the existence and regularity of solutions of linear pseudo-differential equations. Enseign. Math. 17 (1971), 99–163.

    MathSciNet  MATH  Google Scholar 

  9. L. Hörmander: The Cauchy problem for differential equations with double characteristics. J. Analyse Math. 32 (1977), 118–196.

    Article  MathSciNet  MATH  Google Scholar 

  10. L. Hörmander: The analysis of linear partial differential operators III, IV. Grundlehren Math. Wiss. 274, Springer, Berlin, 1985.

    MATH  Google Scholar 

  11. V.Ja. Ivrii, V.M. Petkov: Necessary conditions for the Cauchy problem for nonstrictly hyperbolic equations to be well-posed. Russian Math. Surveys 29 (1974), 1–70.

    Article  MathSciNet  MATH  Google Scholar 

  12. V.Ja. Ivrii: Sufficient conditions for regular and completely regular hyperbolicity. Trans. Moscow. Math. Soc. 33 (1978), 1–65.

    MathSciNet  MATH  Google Scholar 

  13. V.Ja. Ivrii: The well posedness of the Cauchy problem for non-strictly hyperbolic operators, III: The energy integral. Trans. Moscow Math. Soc. 34 (1978), 149–168.

    Google Scholar 

  14. V.Ja. Ivrii: Wave fronts of solutions of certain pseudo-differential equations. Trans. Moscow Math. Soc. 39 (1981), 49–86.

    MathSciNet  Google Scholar 

  15. V.Ja. Ivrii: Wave fronts of solutions of certain hyperbolic pseudo-differential equations. Trans. Moscow Math. Soc. 39 (1981), 87–119.

    Google Scholar 

  16. N. Iwasaki: The Cauchy problem for effectively hyperbolic equations (a special case). J. Math. Kyoto Univ. 23 (1983), 503–562.

    MathSciNet  MATH  Google Scholar 

  17. N. Iwasaki: The Cauchy problem for effectively hyperbolic equations (a standard type). Publ. RIMS Kyoto Univ. 20 (1984), 551–592.

    Article  MathSciNet  MATH  Google Scholar 

  18. N. Iwasaki: The Cauchy problem for effectively hyperbolic equations (general case). J. Math. Kyoto Univ. 25 (1985), 727–743.

    MathSciNet  MATH  Google Scholar 

  19. N. Iwasaki: The Cauchy problem for effectively hyperbolic equations (remarks). In Hyperbolic equations and related topics, S. Mizohata ed., Kinokuniya, Tokyo, 1986, pp. 89–100.

    Google Scholar 

  20. G. Komatsu, T. Nishitani: Continuation of bicharacteristics for effectively hyperbolic operators. Proc. Japan Acad. 65 (1989), 109–112.

    Article  MathSciNet  MATH  Google Scholar 

  21. R.B. Melrose: Equivalence of glancing hypersurfaces. Inventiones Math. 37 (1976), 165–191.

    Article  MathSciNet  MATH  Google Scholar 

  22. R.B. Melrose: The Cauchy problem for effectively hyperbolic operators. Hokkaido Math. J. 12 (1983), 371–391.

    MathSciNet  MATH  Google Scholar 

  23. T. Nishitani: Sur les opérateurs fortement hyperboliques qui ne dépendent que du temps. Sur l'inégalité d'énergie pour une classe d'opérateurs pseudo-différentiels fortement hyperbolyques. Séminaires sur les équations aux dérivées partielles hyperboliques et holomorphes, J. Vaillant, 1981–1982.

    Google Scholar 

  24. T. Nishitani: On the finite propagation speed of wave front sets for effectively hyperbolic operators. Sci. Rep. College Gen. Ed. Osaka Univ. 32:1 (1983), 1–7.

    MathSciNet  MATH  Google Scholar 

  25. T. Nishitani: On wave front sets of solutions for effectively hyperbolic operators. Sci. Rep. College Gen. Ed. Osaka Univ. 32:2 (1983), 1–7.

    MathSciNet  MATH  Google Scholar 

  26. T. Nishitani: A note on reduced forms of effectively hyperbolic operators and energy integrals. Osaka J. Math. 21 (1984), 643–650.

    MathSciNet  MATH  Google Scholar 

  27. T. Nishitani: Local energy integrals for effectively hyperbolic operators, I, II. J. Math. Kyoto Univ. 24 (1984), 623–658, 659–666.

    MathSciNet  MATH  Google Scholar 

  28. T. Nishitani: The Cauchy problem for effectively hyperbolic operators. In Non-linear variational problems, A. Marino, L. Modica, S. Spagnolo and M. De Giovanni eds., Res. Notes in Math. 127, Pitman, London, 1985, pp. 9–23.

    Google Scholar 

  29. T. Nishitani: Microlocal energy estimates for hyperbolic operators with double characteristics. In Hyperbolic equations and related topics, S. Mizohata ed., Kinokuniya, Tokyo, 1986, pp. 235–255.

    Google Scholar 

  30. T. Nishitani: Systèmes effectivement hyperboliques. In Calcul d'opérateurs et fronts d'ondes, J. Vaillant ed., Hermann, Paris, 1988, pp. 108–132.

    Google Scholar 

  31. T. Nishitani: Involutive system of effectively hyperbolic operators. Publ. RIMS Kyoto Univ. 24 (1988), 451–494.

    Article  MathSciNet  MATH  Google Scholar 

  32. D.L. Russell: Numerical solution of singular initial value problems. SIAM J. Numer. Anal. 7 (1970), 399–417.

    Article  MathSciNet  Google Scholar 

  33. S. Wakabayashi: Singularities of solutions of the Cauchy problem for symmetric hyperbolic systems. Comm. Partial Differential Equations 9 (1984), 1147–1177.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, College of General Education Osaka University, Toyonaka, 560, Osaka, Japan

    Tatsuo Nishitani

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  1. Tatsuo Nishitani
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© 1991 Springer-Verlag

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Nishitani, T. (1991). The effectively hyperbolic Cauchy problem. In: The Hyperbolic Cauchy Problem. Lecture Notes in Mathematics, vol 1505. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090884

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  • DOI: https://doi.org/10.1007/BFb0090884

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55018-1

  • Online ISBN: 978-3-540-46655-0

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