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Ill-posedness of a quasilinear wave equation in two dimensions for data in $H^{{7}/{4}}$

Gaspard Ohlmann

Vol. 5 (2023), No. 3, 507–540
DOI: 10.2140/paa.2023.5.507
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 11 4 in our situation), the equation is well-posed in Hs × Hs1. 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 H114(ln H)β × H74(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
Authors
Gaspard Ohlmann
University of Basel
Basel
Switzerland