Archive for Rational Mechanics and Analysis

, Volume 217, Issue 1, pp 113–154 | Cite as

Hyperbolic Second Order Equations with Non-Regular Time Dependent Coefficients

  • Claudia Garetto
  • Michael RuzhanskyEmail author
Open Access


In this paper we study weakly hyperbolic second order equations with time dependent irregular coefficients. This means assuming that the coefficients are less regular than Hölder. The characteristic roots are also allowed to have multiplicities. For such equations, we describe the notion of a ‘very weak solution’ adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifiers of coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or to ultradistributional solutions under conditions when such solutions also exist. In concrete applications, the dependence on the regularising parameter can be traced explicitly.


Weak Solution Cauchy Problem Hyperbolic Equation Energy Estimate Cauchy Data 
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© The Author(s) 2014

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.Department of Mathematical SciencesLoughborough UniversityLoughboroughUK
  2. 2.Department of MathematicsImperial College LondonLondonUK

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