Ill-Posed Cauchy Problem, Revisited

  • Tatsuo Nishitani
Part of the Lecture Notes in Mathematics book series (LNM, volume 2202)


In Chap.  6 we exhibited a second order differential operator with polynomial coefficients for which the Cauchy problem is C ill-posed even though the Levi condition is satisfied. The Levi condition would be the most strict condition that one can impose on lower order terms on the double characteristics as far as we know. In this chapter we confirm this by proving that the Cauchy problem for this operator is ill-posed in the Gevrey class of order grater than 6 for any lower order term. In particular the Cauchy problem is C ill-posed for any lower order term. This phenomenon never occurs in the case of one spatial dimension. In the last section we give an example of second order differential operator of spectral type 1 on Σ, which shows that the IPH condition is not sufficient in general for the Cauchy problem to be C well-posed.


  1. 14.
    F. Colombini, S. Spagnolo, An example of a weakly hyperbolic Cauchy problem not well posed in C . Acta Math. 148, 243–253 (1982)MathSciNetCrossRefMATHGoogle Scholar
  2. 32.
    L. Hörmander, The Cauchy problem for differential equations with double characteristics. J. Anal. Math. 32, 118–196 (1977)MathSciNetCrossRefMATHGoogle Scholar
  3. 36.
    V.Ja. Ivrii, Correctness of the Cauchy problem in Gevrey classes for nonstrictly hyperbolic operators. Uspehi Math. USSR Sbornik 25, 365–387 (1975)CrossRefGoogle Scholar
  4. 38.
    V.Ja. Ivrii, Conditions for correctness in Gevrey classes of the Cauchy problem for hyperbolic operators with characteristics of variable multiplicity. Sib. Math. J. 17, 422–435 (1976)Google Scholar
  5. 39.
    V.Ja. Ivrii, Conditions for correctness in Gevrey classes of the Cauchy problem for not strictly hyperbolic operators. Sib. Math. J. 17, 921–931 (1976)Google Scholar
  6. 43.
    V.Ja. Ivrii, V.M. Petkov, Necessary conditions for the correctness of the Cauchy problem for non-strictly hyperbolic equations. Uspehi Mat. Nauk 29, 3–70 (1974); English transl., Russ. Math. Surv. 29, 1–70 (1974)Google Scholar
  7. 56.
    J. Leray, Y. Ohya, Équations et systèmes non-linéaires, hyperboliques nonstrict. Math. Ann. 170, 167–205 (1967)MathSciNetCrossRefMATHGoogle Scholar
  8. 66.
    S. Mizohata, The Theory of Partial Differential Equations (Cambridge University Press, Cambridge, 1973)MATHGoogle Scholar
  9. 70.
    T. Nishitani, The Cauchy problem for weakly hyperbolic equations of second order. Commun. Partial Differ. Equ. 5, 1273–1296 (1980)MathSciNetCrossRefMATHGoogle Scholar
  10. 81.
    T. Nishitani, On Gevrey well-posedness of the Cauchy problem for some noneffectively hyperbolic operators, in Advances in Phase Space Analysis of PDEs. Progress in Nonlinear Differential Equations and Their Application, vol. 78 (Birkhäuser, Boston, 2009), pp. 217–233Google Scholar
  11. 89.
    C. Parenti, A. Parmeggiani, On the Cauchy problem for hyperbolic operators with double characteristics. Commun. Partial Differ. Equ. 34, 837–888 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 92.
    I.G. Petrovsky, Lectures on Partial Differential Equations (Interscience, New York, 1950)MATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Tatsuo Nishitani
    • 1
  1. 1.Department of MathematicsOsaka UniversityToyonakaJapan

Personalised recommendations