Math 6410 • ODE

Consider the wave equation in three spatial dimensions
\[
  u_{tt}-\Delta u=0 \quad \text{in } \R^3\times(0,\infty).
\]
\begin{enumerate}
\item[(a)] Explain the concept of domain of dependence for solutions of the wave
equation, and describe what this has to do with finite propagation speed.

\item[(b)] Sketch a proof, either using a representation formula or energy methods, that
solutions \(u\in C^2\) of the wave equation have an explicitly defined domain of
dependence.
\end{enumerate}