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On the Cauchy Problem for a Weakly Hyperbolic Operator: An Intermediate Case Between Effective Hyperbolicity and Levi Condition

  • Ferruccio Colombini
  • Mariagrazia Di Flaviano
  • Tatsuo Nishitani
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 52)

Abstract

We are interested in the Cauchy problem for second order hyperbolic operators of the form
$$ \left\{ {\begin{array}{*{20}{c}} {P\left( {t,x,{\partial _t},{\partial _x}} \right)u\left( {t,x} \right) = f\left( {t,x} \right),} \\ {u\left( {0,x} \right) = {u_0}\left( x \right),{\partial _t}u\left( {0,x} \right) = {u_1}\left( x \right)} \end{array}} \right.$$
(1.1)
on \( \left[ {0,T} \right] \times {R^n}\) where
$$ P\left( {t,x,{\partial _t},{\partial _x}} \right) = {P_2}\left( {t,{\partial _t},{\partial _x}} \right) + {P_1}\left( {t,{\partial _t},{\partial _x}} \right) + c\left( {t,x} \right),{P_2}\left( {t,{\partial _t},{\partial _x}} \right) = \partial _t^2 - \sum\nolimits_{i,j = 1}^n {{a_{ij}}\left( t \right){\partial _{{x_i}}}{\partial _{{x_j}}}},{P_1}\left( {t,{\partial _t},{\partial _x}} \right) = \sum\nolimits_{j = 1}^n {{b_j}\left( t \right){\partial_{{x_j}}} + {b_0}\left( t \right){\partial _t}} .$$
(1.2)
We assume that \( {a_{ij}} \in {C^\infty }\left( {\left[ {0,T} \right]} \right){b_j} \in C\left( {\left[ {0,T} \right]} \right)c \in C\left( {\left[ {0,T} \right];{C^\infty }\left( {{R^n}} \right)} \right);\) moreover
$$ a\left( {t,\xi } \right) = \sum\limits_{i,j = 1}^n {{a_{ij}}\left( t \right){\xi _i}{\xi _j} \geqslant 0,\forall \xi \in {R^n}},t \in \left[ {0,T} \right].$$
(1.3)
This problem has been extensively studied by many authors, starting from the work of E.E. Levi [6], who, for the first time, pointed out that the Cauchy problem for the operator
$$ \partial _t^2 - {\partial _x}$$
(1.4)
is solvable neither in Cnor in the Gevrey classes γsfor s > 2.

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References

  1. 1.
    F. Colombini, H. Ishida and N. Orrù, On the Cauchy problem for finitely degenerate hyperbolic equations of second orderArk. Mat.38 (2000), 223–230.Google Scholar
  2. 2.
    F. Colombini, E. Jannelli and S. Spagnolo, Well-posedness in the Gevrey classes of the Cauchy problem for a non-strictly hyperbolic equation with coefficients depending on timeAnn. Scuola Norm. Sup. Pisa 10(1983), 291–312.MathSciNetzbMATHGoogle Scholar
  3. 3.
    F. Colombini and T. Nishitani, Two by two Strongly Hyperbolic Systems and Gevrey ClassesAnn. Univ. Ferrara Sez. VII Sc. Mat. Suppl.45 (1999), 79–108.MathSciNetzbMATHGoogle Scholar
  4. 4.
    F. Colombini and S. Spagnolo, An example of a weakly hyperbolic Cauchy problem not well posed inC°O Acta. Math.148 (1982), 243–253.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    J. HadamardLe problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliquesHermann et Cie, Paris, 1932.Google Scholar
  6. 6.
    E.E. Levi, Sul problema di CauchyRend. Reale Acad. Lincei16 (1907), 105–112.zbMATHGoogle Scholar
  7. 7.
    V. Ya. Ivrii and V. M. Petkov, Necessary conditions for the well posedness of the Cauchy problem for non strictly hyperbolic equationsUsp. Mat. Nauk29, (1974), 3–70, (Russian). English transl.:Russ. Math. Surv.29, (1974), 1–70.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    V. Ya. Ivrii, Cauchy problem conditions for hyperbolic operators with characteristics of variable multiplicity for Gevrey classesSibirsk. Mat. Zh.17, (1976), 1256–1270, (Russian). English transl.:Siberian Math. J.17, (1976), 921–931.Google Scholar
  9. 9.
    K. Kajitani, The well-posed Cauchy problem for hyperbolic operators, Séminaire Jean Vaillant 1989.Google Scholar
  10. 10.
    T. Nishitani, The effectively hyperbolic Cauchy problem, inThe Hyperbolic Cauchy Problem(by K. Kajitani and T. Nishitani), Lecture Notes in Math. 1505, pp. 71–167, Springer-Verlag, Berlin-Heidelberg, 1991.Google Scholar
  11. 11.
    N. Orrù, On a weakly hyperbolic equation with a term of order zeroAnn. Fac. Sci. Toul.6, 3 (1997), 525–534.MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Ferruccio Colombini
    • 1
  • Mariagrazia Di Flaviano
    • 2
  • Tatsuo Nishitani
    • 3
  1. 1.Dipartimento di Matematica Università di Pisa Via F.Buonarroti 2Italy
  2. 2.Università dell’AquilaDipartimento di Matematica Pura ed ApplicataCoppitoItaly
  3. 3.Department of MathematicsOsaka UniversityToyonaka OsakaJapan

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