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A generalized Levi condition for weakly hyperbolic Cauchy problems with coefficients low regular in time and smooth in space

  • Daniel LorenzEmail author
  • Michael Reissig
Article
  • 35 Downloads

Abstract

We consider the Cauchy problem for weakly hyperbolic m-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that in general one has to impose Levi conditions to get \(C^\infty \) or Gevrey well-posedness even if the coefficients are smooth. We use moduli of continuity to describe the regularity of the coefficients with respect to time, weight sequences for the characterization of their regularity with respect to space and weight functions to define the solution spaces. Furthermore, we propose a generalized Levi condition that models the influence of multiple characteristics more freely. We establish sufficient conditions for the well-posedness of the Cauchy problem, that link the Levi condition as well as the modulus of continuity and the weight sequence of the coefficients to the weight function of the solution space. Additionally, we obtain that the influences of the Levi condition and the low regularity of coefficients on the weight function of the solution space are independent of each other.

Keywords

Weakly hyperbolic Cauchy problem Levi condition Modulus of continuity Weight sequence Weight function 

Mathematics Subject Classification

35S05 35L30 47G30 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Applied Analysis, Faculty of Mathematics and Computer ScienceTU Bergakademie FreibergFreibergGermany

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