Skip to main content
SpringerLink
Log in

The effectively hyperbolic Cauchy problem

  • Chapter
  • First Online:
The Hyperbolic Cauchy Problem

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

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 29.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 39.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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 

Download references

Author information

Authors and Affiliations

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

    Tatsuo Nishitani

Authors
  1. Tatsuo Nishitani
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag

About this chapter

Cite this chapter

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

Download citation

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

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55018-1

  • Online ISBN: 978-3-540-46655-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation