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Hyperbolic second order equations with non-regular time dependent coefficients

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Abstract
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 Holder. 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.
Keywords
CAUCHY-PROBLEM, DISCONTINUOUS COEFFICIENTS, WELL-POSEDNESS, REGULARITY, OPERATORS, SYSTEMS

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MLA
Garetto, Claudia, and Michael Ruzhansky. “Hyperbolic Second Order Equations with Non-Regular Time Dependent Coefficients.” ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, vol. 217, no. 1, 2014, pp. 113–54, doi:10.1007/s00205-014-0830-1.
APA
Garetto, C., & Ruzhansky, M. (2014). Hyperbolic second order equations with non-regular time dependent coefficients. ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 217(1), 113–154. https://doi.org/10.1007/s00205-014-0830-1
Chicago author-date
Garetto, Claudia, and Michael Ruzhansky. 2014. “Hyperbolic Second Order Equations with Non-Regular Time Dependent Coefficients.” ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS 217 (1): 113–54. https://doi.org/10.1007/s00205-014-0830-1.
Chicago author-date (all authors)
Garetto, Claudia, and Michael Ruzhansky. 2014. “Hyperbolic Second Order Equations with Non-Regular Time Dependent Coefficients.” ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS 217 (1): 113–154. doi:10.1007/s00205-014-0830-1.
Vancouver
1.
Garetto C, Ruzhansky M. Hyperbolic second order equations with non-regular time dependent coefficients. ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS. 2014;217(1):113–54.
IEEE
[1]
C. Garetto and M. Ruzhansky, “Hyperbolic second order equations with non-regular time dependent coefficients,” ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, vol. 217, no. 1, pp. 113–154, 2014.
@article{8585354,
  abstract     = {{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 Holder. 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.}},
  author       = {{Garetto, Claudia and Ruzhansky, Michael}},
  issn         = {{0003-9527}},
  journal      = {{ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS}},
  keywords     = {{CAUCHY-PROBLEM,DISCONTINUOUS COEFFICIENTS,WELL-POSEDNESS,REGULARITY,OPERATORS,SYSTEMS}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{113--154}},
  title        = {{Hyperbolic second order equations with non-regular time dependent coefficients}},
  url          = {{http://doi.org/10.1007/s00205-014-0830-1}},
  volume       = {{217}},
  year         = {{2014}},
}

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