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### Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I. Well-posedness

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 Title: Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I. Well-posedness Authors: Garetto, CJäh, CRuzhansky, M Item Type: Journal Article Abstract: In this paper we analyse the well-posedness of the Cauchy problem for a rather general class of hyperbolic systems with space-time dependent coefficients and with multiple characteristics of variable multiplicity. First, we establish a well-posedness result in anisotropic Sobolev spaces for systems with upper triangular principal part under interesting natural conditions on the orders of lower order terms below the diagonal. Namely, the terms below the diagonal at a distance $k$ to it must be of order $-k$. This setting also allows for the Jordan block structure in the system. Second, we give conditions for the Schur type triangularisation of general systems with variable coefficients for reducing them to the form with an upper triangular principal part for which the first result can be applied. We give explicit details for the appearing conditions and constructions for $2\times 2$ and $3\times 3$ systems, complemented by several examples. Issue Date: 22-Mar-2018 Date of Acceptance: 16-Mar-2018 URI: http://hdl.handle.net/10044/1/58418 DOI: https://dx.doi.org/10.1007/s00208-018-1672-1 ISSN: 0025-5831 Publisher: Springer Verlag Journal / Book Title: Mathematische Annalen Copyright Statement: © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Sponsor/Funder: The Leverhulme TrustEngineering & Physical Science Research Council (EPSRC) Funder's Grant Number: RPG-2017-151EP/R003025/1 Keywords: math.APPrimary 35L45, Secondary 46E350101 Pure MathematicsGeneral Mathematics Publication Status: Published online Appears in Collections: Pure MathematicsMathematicsFaculty of Natural Sciences