On the stability of the ill-posedness of a quasi-linear wave equation in two dimensions with initial data in $H^{7/4} (\ln H^{-\beta})$

TL;DR

This paper investigates the persistent ill-posedness of a quasi-linear wave equation in two dimensions with initial data in a specific Sobolev space, demonstrating that blow-up phenomena occur under various modifications.

Contribution

It extends previous work by showing that ill-posedness persists even with modifications to the equation and initial data dependencies, including perturbations.

Findings

Blow-up phenomena occur in modified equations.

Ill-posedness persists without explicit formulas.

Pathological behavior remains with initial data perturbations.

Abstract

This article is the continuation of \cite{ohlmann2021illposedness} where we exhibited the ill-posedness of a quasi-linear wave equation in dimension $2$ for initial data in $H^{7/4}(\ln H^{7/4})$. Here, we look at modifications of the equation and show that the blow-up phenomenon still occurs. First, we study another equation with the same characteristics but a different underlying ODE. Later, we study an equation where a $x_2$ dependency is introduced. The latter case constitutes the main contribution of this paper, as we are able to show that the solution still behaves pathologically without having an explicit formula for either the characteristics or the values of the solution. Finally, we study the case where a perturbation of the initial data is introduced.