Math 6410 • ODE

Consider the scalar first order PDE
\begin{equation}
\begin{aligned}
  uu_x+u_y&=0,\\
  u(x,0)&=h(x).
\end{aligned}
\tag{4}
\end{equation}
\begin{enumerate}
\item[(a)] Define what it means to be a weak solution of (4).
\item[(b)] Suppose that \(u_0>0\) and
\[
  h(x)=
  \begin{cases}
    u_0, & \text{if }x\le0,\\
    u_0(1-x), & \text{if }0<x<1,\\
    0, & \text{if }x\ge1.
  \end{cases}
\]
Show that a shock develops at a finite time and describe the global weak solution.
\end{enumerate}