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

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 $H^{7\mathrm{/}4}(\ln H^{7\mathrm{/}4})$. 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.