Math 6410 • ODE

\begin{enumerate}
\item[(a)] Suppose an initial profile \(u_0(x)\) is monotone with
\(\lim_{x\to-\infty}u_0(x, )=u_L\), \(\lim_{x\to\infty}u_0(x)=u_R\). Under what
conditions on \(u_L\) and \(u_R\) does the solution of Burger's equation
\[
  \frac{\partial u}{\partial t}
  +u\frac{\partial u}{\partial x}=0,
\]
evolve into a shock, and what is its asymptotic speed?
\item[(b)] Consider the viscous Burger's equation
\[
  \frac{\partial u}{\partial t}
  +u\frac{\partial u}{\partial x}
  =\epsilon\frac{\partial^2u}{\partial x^2}.
\]
with \(\epsilon>0\). Under what conditions on \(u_L\) and \(u_R\), where
\(\lim_{x\to-\infty}u(x,t)=u_L\), \(\lim_{x\to\infty}u(x,t)=u_R\) does this equation
have a travelling wave solution, and what is its speed? Verify your result.
\end{enumerate}