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Abstract
We study the ill-posedness of a quasilinear wave equation. It was shown by Smith and Tataru in 2005
that for any
s
> 7
4 (or
1 1
4 in our situation), the
equation is well-posed in
H s
× H s − 1 .
We show a sharpness result by exhibiting a quasilinear wave equation
and an initial data such that the Cauchy problem is ill-posed for in
H 1 1 ∕ 4 ( ln H ) − β
× H 7 ∕ 4 ( ln H ) − β .
Keywords
PDE, quasilinear wave, wave equation, ill-posedness,
well-posedness, Sobolev spaces, analysis, fractional
derivatives
Mathematical Subject Classification
Primary: 35A01, 35A30, 35L05
Secondary: 35A21, 35D30
Milestones
Received: 30 July 2021
Revised: 14 January 2023
Accepted: 12 March 2023
Published: 24 August 2023
© 2023 MSP (Mathematical Sciences
Publishers).