art%3A10.1007%2Fs00205-014-0830-1.pdf (425.62 kB)
Download file

Hyperbolic second order equations with non-regular time dependent coefficients

Download (425.62 kB)
journal contribution
posted on 15.04.2015, 10:55 by Claudia Garetto, Michael Ruzhansky
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.

Funding

M. Ruzhansky was supported by the EPSRC Leadership Fellowship EP/G007233/1 and by EPSRC Grant EP/K039407/1

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Archive for Rational Mechanics and Analysis

Volume

217

Issue

1

Pages

113 - 154

Citation

GARETTO, C. and RUZHANSKY, M., 2015. Hyperbolic second order equations with non-regular time dependent coefficients. Archive for Rational Mechanics and Analysis, 217(1), pp. 113-154.

Publisher

© The Author(s). This article is published with open access at Springerlink.com

Version

VoR (Version of Record)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution 3.0 Unported (CC BY 3.0) licence. Full details of this licence are available at: http://creativecommons.org/licenses/by/3.0/

Acceptance date

02/12/2014

Publication date

2014-12-20

Notes

This 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.

ISSN

0003-9527

eISSN

1432-0673

Language

en

Usage metrics

Keywords

Licence

Exports