(a) Solve by using the method of characteristics

$$x_{1} \frac{\partial}{\partial x_{1}} u+2 x_{2} \frac{\partial}{\partial x_{2}} u=5 u, \quad u\left(x_{1}, 1\right)=g\left(x_{1}\right),$$

where $g: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. What is the maximal domain in $\mathbb{R}^{2}$ in which $u$ is a solution of the Cauchy problem?

(b) Prove that the function

$$u(x, t)=\left\{\begin{array}{cl} 0, & x<0, t>0 \\ x / t, & 0<x<t, t>0 \\ 1, & x>t>0 \end{array}\right.$$

is a weak solution of the Burgers equation

$$\frac{\partial}{\partial t} u+\frac{1}{2} \frac{\partial}{\partial x} u^{2}=0, \quad x \in \mathbb{R}, t>0$$

with initial data

$$u(x, 0)= \begin{cases}0, & x<0 \\ 1, & x>0\end{cases}$$

(c) Let $u=u(x, t), x \in \mathbb{R}, t>0$ be a piecewise $C^{1}$-function with a jump discontinuity along the curve

$$\Gamma: x=s(t)$$

and let $u$ solve the Burgers equation $(*)$ on both sides of $\Gamma$. Prove that $u$ is a weak solution of (1) if and only if

$$\dot{s}(t)=\frac{1}{2}\left(u_{l}(t)+u_{r}(t)\right)$$

holds, where $u_{l}(t), u_{r}(t)$ are the one-sided limits

$$u_{l}(t)=\lim _{x \nearrow_{s}(t)^{-}} u(x, t), \quad u_{r}(t)=\lim _{x \searrow s(t)^{+}} u(x, t)$$

[Hint: Multiply the equation by a test function $\phi \in C_{0}^{\infty}(\mathbb{R} \times[0, \infty))$, split the integral appropriately and integrate by parts. Consider how the unit normal vector along $\Gamma$ can be expressed in terms of $\dot{s}$.]