caa a22 4500 605515433 CHVBK 20210128100709.0 cr unu---uuuuu 210128e20150701xx s 000 0 eng 10.1007/s00205-014-0830-1 doi (NATIONALLICENCE)springer-10.1007/s00205-014-0830-1 Hyperbolic Second Order Equations with Non-Regular Time Dependent Coefficients [Elektronische Daten] [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. The Author(s), 2014 Garetto Claudia Department of Mathematical Sciences, Loughborough University, LE11 3TU, Loughborough, Leicestershire, UK aut Ruzhansky Michael Department of Mathematics, Imperial College London, 180 Queen's Gate, SW7 2AZ, London, UK aut Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 217/1(2015-07-01), 113-154 0003-9527 217:1<113 2015 217 205 https://doi.org/10.1007/s00205-014-0830-1 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00205-014-0830-1 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Garetto Claudia Department of Mathematical Sciences, Loughborough University, LE11 3TU, Loughborough, Leicestershire, UK aut NATIONALLICENCE 700 1- Ruzhansky Michael Department of Mathematics, Imperial College London, 180 Queen's Gate, SW7 2AZ, London, UK aut NATIONALLICENCE 773 0- Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 217/1(2015-07-01), 113-154 0003-9527 217:1<113 2015 217 205