Finite vs infinite derivative loss for abstract wave equations with singular time-dependent propagation speed

We consider an abstract wave equation with a propagation speed that depends only on time. We investigate well-posedness results with finite derivative loss in the case where the propagation speed is smooth for positive times, but potentially singular at the initial time. We prove that solutions exhi...

Full description

Saved in:
Bibliographic Details
Published in:Bulletin des sciences mathématiques 2021-02, Vol.166, p.102918, Article 102918
Main Authors: Ghisi, Marina, Gobbino, Massimo
Format: Article
Language:English
Subjects:
Baire category
Finite derivative loss
Infinite derivative loss
Linear hyperbolic equation
Residual set
Wave equation
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We consider an abstract wave equation with a propagation speed that depends only on time. We investigate well-posedness results with finite derivative loss in the case where the propagation speed is smooth for positive times, but potentially singular at the initial time. We prove that solutions exhibit a finite derivative loss under a family of conditions that involve the blow up rate of the first and second derivative of the propagation speed, in the spirit that the weaker is the requirement on the first derivative, the stronger is the requirement on the second derivative. Our family of conditions interpolates between the two limit cases that were already known in the literature. We also provide the counterexamples that show that, as soon as our conditions fail, solutions can exhibit an infinite derivative loss. The existence of such pathologies was an open problem even in the two extreme cases.
ISSN:0007-4497
1952-4773
DOI:10.1016/j.bulsci.2020.102918