Math 6410 • ODE

Consider the partial differential equation defined for \(-\infty<x<\infty\)
\[
  \frac{\partial u}{\partial t}
  -\frac{\partial^2u}{\partial x^2}
  =\chi(x)f(u),
  \qquad
  \chi(x)=
  \begin{cases}
    0 & 0<x<L,\\
    1 & \text{elsewhere}
  \end{cases}
\]
with \(f(u)\) a bistable function, \(f(0)=f(a)=f(1)=0\), \(0<a<1\),
\(f'(0)<0\), \(f'(1)<0\) and
\[
  \int_0^1 f(u)\dd u>0.
\]
\begin{enumerate}
\item[(a)] Prove that for all \(L>0\) sufficiently large, there is a standing, monotone
increasing \(C^1\) profile with limiting behavior \(u(-\infty)=0\), \(u(\infty)=1\).
\item[(b)] State a comparison theorem that can be used to show that if \(L\) is
sufficiently large, the region \(0<x<L\) acts as a ``blocking region'' to propagation
for this equation.
\end{enumerate}