Math 6410 • ODE

Consider the equation
\[
  a(x,y)u_x+b(x,y)u_y=c(x,y,u)
  \tag{8.1}
\]
where \(a\), \(b\) and \(c\) are suitably smooth.
\begin{enumerate}
\item Define what is meant by an integral surface for (8.1).
\item Write down the characteristic equations for (8.1) and show how they can be used
to construct an integral surface of (8.1) that contains the curve
\[
  x=\varphi(s)\qquad y=\psi(s)\qquad u=\eta(s),
\]
where \(\varphi,\psi\) and \(\eta\) are \(C^1\).
\item What additional conditions, if any, must be placed on the data to guarantee a
locally unique solution? Explain carefully.
\item Consider the equation
\[
  u_t+uu_x=0
  \tag{8.2}
\]
with initial data given by
\[
  u(x,0)=u_0(x)=
  \begin{cases}
    1, & x\le0,\\
    1-x, & x\in(0,1),\\
    0, & x\ge1.
  \end{cases}
  \tag{8.3}
\]
Does (8.2)--(8.3) have a unique classical solution for \(t\ge0\)? A unique weak solution
for \(t\ge0\)? Explain.
\end{enumerate}