Math 6410 • ODE

\textbf{Wave Propagation.}
Consider the equation
\[
  u_{tt}-u_{xx}+\lambda u=0,\quad x\in\R,\quad t>0,
\]
where \(\lambda>0\) is fixed.
\begin{enumerate}
\item[(a)] Explain what is meant by a dispersive solution, and find a dispersion relation
for the equation.
\item[(b)] Define an ``energy'' functional \(E(t)\) for this equation and show that each
solution \(u(x,t)\) with finite initial energy conserves energy. State conditions under
which uniqueness of solutions holds.
\end{enumerate}