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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Precise finite speed and uniqueness in the Cauchy problem for symmetrizable hyperbolic systems


Author: Jeffrey Rauch
Journal: Trans. Amer. Math. Soc. 363 (2011), 1161-1182
MSC (2000): Primary 35L45; Secondary 35S10
Published electronically: October 21, 2010
MathSciNet review: 2737261
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Abstract: Precise finite speed, in the sense of the domain of influence being a subset of the union of influence curves through the support of the initial data, is proved for hyperbolic systems symmetrized by pseudodifferential operators in the spatial variables. From this, uniqueness in the Cauchy problem at spacelike hypersurfaces is derived by a Hölmgren style duality argument. Sharp finite speed is derived from an estimate for propagation in each direction. Propagation in a fixed direction is proved by regularizing the problem in the orthogonal directions. Uniform estimates for the regularized equations are proved using pseudodifferential techniques of Beals-Fefferman type.


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Additional Information

Jeffrey Rauch
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: rauch@umich.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05047-0
PII: S 0002-9947(2010)05047-0
Keywords: Finite speed, domain of influence, symmetrizable hyperbolic system, uniqueness in the Cauchy problem
Received by editor(s): July 2, 2008
Published electronically: October 21, 2010
Additional Notes: This research was partially supported by the National Science Foundation under grant NSF DMS 0405899
Article copyright: © Copyright 2010 American Mathematical Society