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The Cauchy problem for a class of Kovalevskian pseudo-differential operators

Authors: Rossella Agliardi and Massimo Cicognani
Journal: Proc. Amer. Math. Soc. 132 (2004), 841-845
MSC (2000): Primary 35G10, 35L30
Published electronically: August 19, 2003
MathSciNet review: 2019963
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Abstract: We prove the $H^{\infty}$ well-posedness of the forward Cauchy problem for a pseudo-differential operator $P$ of order $m\geq 2$ with the Log-Lipschitz continuous symbol in the time variable. The characteristic roots $\lambda_k$ of $P$ are distinct and satisfy the necessary Lax-Mizohata condition Im $\lambda_k\geq 0$. The Log-Lipschitz regularity has been tested as the optimal one for $H^{\infty}$ well-posedness in the case of second-order hyperbolic operators. Our main aim is to present a simple proof which needs only a little of the basic calculus of standard pseudo-differential operators.

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Additional Information

Rossella Agliardi
Affiliation: University of Ferrara, via Machiavelli 35, 44100 Ferrara, Italy

Massimo Cicognani
Affiliation: University of Bologna, via Genova 181, 47023 Cesena, Italy

Keywords: Strictly hyperbolic operators, energy estimates, Log-Lipschitz continuity
Received by editor(s): September 30, 2002
Received by editor(s) in revised form: November 5, 2002
Published electronically: August 19, 2003
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2003 American Mathematical Society

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