Math 6410 • ODE

Consider the one-dimensional parabolic PDE
\[
  \frac{\partial u}{\partial t}
  +\frac{\partial}{\partial x}\left(\frac{u^n}{n}\right)
  =\epsilon\frac{\partial^2u}{\partial x^2}.
\]
\begin{enumerate}
\item[(a)] Show that a traveling wave solution \(u(x,t)=U(z)\) with
\(z=(x-Vt)/\epsilon\) satisfies
\[
  \frac{dU}{dz}=\frac{U^n}{n}-VU+\text{constant},
\]
and deduce that
\[
  V=\frac{[U^n/n]_{-\infty}^{\infty}}{[U]_{-\infty}^{\infty}}.
\]
\item[(b)] Discuss how the traveling wave solution relates to shock solutions of the
quasilinear equation obtained by setting \(\epsilon=0\). Also show that, when \(n=2\),
\(U\) can only tend to \(U(\pm\infty)\) as \(z\to\pm\infty\) if \(dU/dz<0\).
\end{enumerate}