Consider the equation

$$x_{2} \frac{\partial u}{\partial x_{1}}-x_{1} \frac{\partial u}{\partial x_{2}}+a \frac{\partial u}{\partial x_{3}}=u,$$

where $a \in \mathbb{R}$, to be solved for $u=u\left(x_{1}, x_{2}, x_{3}\right)$. State clearly what it means for a hypersurface

$$S_{\phi}=\left\{\left(x_{1}, x_{2}, x_{3}\right): \phi\left(x_{1}, x_{2}, x_{3}\right)=0\right\},$$

defined by a $C^{1}$ function $\phi$, to be non-characteristic for $(*)$. Does the non-characteristic condition hold when $\phi\left(x_{1}, x_{2}, x_{3}\right)=x_{3}$ ?

Solve $(*)$ for $a>0$ with initial condition $u\left(x_{1}, x_{2}, 0\right)=f\left(x_{1}, x_{2}\right)$ where $f \in C^{1}\left(\mathbb{R}^{2}\right)$. For the case $f\left(x_{1}, x_{2}\right)=x_{1}^{2}+x_{2}^{2}$ discuss the limiting behaviour as $a \rightarrow 0_{+}$.