(a) State the local existence theorem of a classical solution of the Cauchy problem

$$\begin{aligned} &a\left(x_{1}, x_{2}, u\right) \frac{\partial u}{\partial x_{1}}+b\left(x_{1}, x_{2}, u\right) \frac{\partial u}{\partial x_{2}}=c\left(x_{1}, x_{2}, u\right) \\ &\left.u\right|_{\Gamma}=u_{0} \end{aligned}$$

where $\Gamma$ is a smooth curve in $\mathbb{R}^{2}$.

(b) Solve, by using the method of characteristics,

$$\begin{aligned} &2 x_{1} \frac{\partial u}{\partial x_{1}}+4 x_{2} \frac{\partial u}{\partial x_{2}}=u^{2} \\ &u\left(x_{1}, 2\right)=h \end{aligned}$$

where $h>0$ is a constant. What is the maximal domain of existence in which $u$ is a solution of the Cauchy problem?