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)


We are interested in the Cauchy problem for second order hyperbolic operators of the form
$$ \left\{ {\begin{array}{*{20}{c}} {P\left( {t,x,{\partial _t},{\partial _x}} \right)u\left( {t,x} \right) = f\left( {t,x} \right),} \\ {u\left( {0,x} \right) = {u_0}\left( x \right),{\partial _t}u\left( {0,x} \right) = {u_1}\left( x \right)} \end{array}} \right.$$
on \( \left[ {0,T} \right] \times {R^n}\) where
$$ P\left( {t,x,{\partial _t},{\partial _x}} \right) = {P_2}\left( {t,{\partial _t},{\partial _x}} \right) + {P_1}\left( {t,{\partial _t},{\partial _x}} \right) + c\left( {t,x} \right),{P_2}\left( {t,{\partial _t},{\partial _x}} \right) = \partial _t^2 - \sum\nolimits_{i,j = 1}^n {{a_{ij}}\left( t \right){\partial _{{x_i}}}{\partial _{{x_j}}}},{P_1}\left( {t,{\partial _t},{\partial _x}} \right) = \sum\nolimits_{j = 1}^n {{b_j}\left( t \right){\partial_{{x_j}}} + {b_0}\left( t \right){\partial _t}} .$$
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
$$ a\left( {t,\xi } \right) = \sum\limits_{i,j = 1}^n {{a_{ij}}\left( t \right){\xi _i}{\xi _j} \geqslant 0,\forall \xi \in {R^n}},t \in \left[ {0,T} \right].$$
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
$$ \partial _t^2 - {\partial _x}$$
is solvable neither in Cnor in the Gevrey classes γsfor s > 2.


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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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