Abstract
We are concerned with analyzing hyperbolic equations with distributional coefficients. We focus on the case of coefficients with jump discontinuities considered earlier by Hurd and Sattinger in their proof of the breakdown of global distributional solutions. Within the framework of Colombeau generalized functions, however, Oberguggenberger showed the existence and uniqueness of a global solution. Within this framework we develop further a microlocal analysis to understand the propagation of singularities of such Colombeau solutions. To achieve this we introduce a refined notion of a wave-front set, extending Hörmander's definition for distributions. We show how the coefficient singularities modify the classical relation of the wave front set of the solution and the characteristic set of the operator, with a generalized notion of characteristic set.
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Hörmann, G., de Hoop, M.V. Microlocal Analysis and Global Solutions of Some Hyperbolic Equations with Discontinuous Coefficients. Acta Applicandae Mathematicae 67, 173–224 (2001). https://doi.org/10.1023/A:1010614332739
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DOI: https://doi.org/10.1023/A:1010614332739