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Transactions of the American Mathematical Society

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On the wave equation with multiplicities and space-dependent irregular coefficients


Author: Claudia Garetto
Journal: Trans. Amer. Math. Soc. 374 (2021), 3131-3176
MSC (2020): Primary 35L05, 35L15
DOI: https://doi.org/10.1090/tran/8319
Published electronically: February 12, 2021
MathSciNet review: 4237945
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Abstract: In this paper we study the well-posedness of the Cauchy problem for a wave equation with multiplicities and space-dependent irregular coefficients. As in Garetto and Ruzhansky [Arch. Ration. Mech. Anal. 217 (2015), pp. 113–154], in order to give a meaningful notion of solution, we employ the notion of very weak solution, which construction is based on a parameter dependent regularisation of the coefficients via mollifiers. We prove that, even with distributional coefficients, a very weak solution exists for our Cauchy problem and it converges to the classical one when the coefficients are smooth. The dependence on the mollifiers of very weak solutions is investigated at the end of the paper in some instructive examples.


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Additional Information

Claudia Garetto
Affiliation: Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, United Kingdom
MR Author ID: 729291
ORCID: 0000-0001-7907-6072
Email: c.garetto@lboro.ac.uk

Keywords: Wave equation, multiplicities, singularities
Received by editor(s): May 26, 2020
Published electronically: February 12, 2021
Article copyright: © Copyright 2021 American Mathematical Society