(a) Solve the initial value problem for the Burgers equation

$$\begin{gathered} u_{t}+\frac{1}{2}\left(u^{2}\right)_{x}=0, \quad x \in \mathbb{R}, t>0 \\ u(x, t=0)=u_{I}(x) \end{gathered}$$

where

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

Use the method of characteristics. What is the maximal time interval in which this (weak) solution is well defined? What is the regularity of this solution?

(b) Apply the method of characteristics to the Burgers equation subject to the initial condition

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

In $\{(x, t) \mid 0<x<t\}$ use the ansatz $u(x, t)=f\left(\frac{x}{t}\right)$ and determine $f$.

(c) Using the method of characteristics show that the initial value problem for the Burgers equation has a classical solution defined for all $t>0$ if $u_{I}$ is continuously differentiable and

$$\frac{d u_{I}}{d x}(x)>0$$

for all $x \in \mathbb{R}$.