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Mathematische Annalen

, Volume 357, Issue 2, pp 401–440

First Online: 05 February 2013Received: 28 May 2012Revised: 24 October 2012

Abstract

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^k\-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^\infty \ 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.

Mathematics Subject ClassificationPrimary 35G10 35L30 Secondary 46F05 The first author supported by the Imperial College Junior Research Fellowship. The second author supported by EPSRC Leadership Fellowship EP-G007233-1.

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Autor: Claudia Garetto - Michael Ruzhansky

Fuente: https://link.springer.com/



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