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Weakly hyperbolic equations with non-analytic coefficients and lower order terms

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journal contribution
posted on 28.07.2014 by Claudia Garetto, Michael Ruzhansky
In this paper we consider weakly hyperbolic equations of higher orders in arbitrary dimensions with time-dependent coefficients and lower order terms. We prove the Gevrey well-posedness of the Cauchy problem under C-regularity of coefficients of the principal part and natural Levi conditions on lower order terms which may be only continuous. In the case of analytic coefficients in the principal part we establish the C well-posedness. The proofs are based on using the quasi-symmetriser for the corresponding companion system and inductions on the order of equation and on the frequency regions. The main novelty compared to the existing literature is the possibility to include lower order terms to the equation (which have been untreatable until now in these problems) as well as considering any space dimensions. We also give results on the ultradistributional and distributional well-posedness of the problem, and we look at new effects for equations with discontinuous lower order terms.

Funding

[C. Garetto was] supported by the Imperial College Junior Research Fellowship. [M. Ruzhansky was] supported by EPSRC Leadership Fellowship EP/G007233/1.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Mathematische Annalen

Volume

357

Issue

2

Pages

401 - 440

Citation

GARETTO, C. and RUZHANSKY, M., 2013. Weakly hyperbolic equations with non-analytic coefficients and lower order terms. Mathematische Annalen, 357 (2), pp. 401-440.

Publisher

Springer (© the authors)

Version

AM (Accepted Manuscript)

Publication date

2013

Notes

The final publication is available at Springer via http://dx.doi.org/10.1007/s00208-013-0910-9.

ISSN

0025-5831

eISSN

1432-1807

Language

en

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