# 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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## 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