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Nonlinear Hyperbolic Equations, Spectral Theory, and Wavelet Transformations pp 157–205Cite as

Birkhäuser

Local Solutions to Quasi-linear Weakly Hyperbolic Differential Equations

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Part of the book series: Operator Theory: Advances and Applications ((APDE,volume 145))

Abstract

The purpose of this paper is to investigate weakly hyperbolic equations with degeneracies in the space and time variables. These degeneracies as well as the sharp Levi conditions of C ∞ type are formulated by means of certain weight functions. For Cauchy problems to such quasi-linear weakly hyperbolic equations, the following subjects are studied: local existence of solutions in Sobolev spaces and C ∞, a blow-up criterion, domains of dependence, and C ∞ regularity. The main tools are the transformation of the higher-order equation to a first-order system, a calculus for pseudodifferential operators with non-smooth symbols, and a generalization of Gronwall’s lemma to differential inequalities with a singular coefficient.

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Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Freiberg University of Mining and Technology, Agricola-Straße 1, Freiberg, D-09596, Germany

    Michael Dreher

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  1. Michael Dreher
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Editor information

Editors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Bonn, Bonn, 53115, Germany

    Sergio Albeverio

  2. Institut für Mathematik, Technische Universität Clausthal, Clausthal-Zellerfeld, 38678, Germany

    Michael Demuth

  3. Institut für Mathematik, Universität Potsdam, Potsdam, 14415, Germany

    Elmar Schrohe  & Bert-Wolfgang Schulze  & 

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Dreher, M. (2003). Local Solutions to Quasi-linear Weakly Hyperbolic Differential Equations. In: Albeverio, S., Demuth, M., Schrohe, E., Schulze, BW. (eds) Nonlinear Hyperbolic Equations, Spectral Theory, and Wavelet Transformations. Operator Theory: Advances and Applications, vol 145. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8073-2_3

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  • DOI: https://doi.org/10.1007/978-3-0348-8073-2_3

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9429-6

  • Online ISBN: 978-3-0348-8073-2

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