Math 6410 • ODE

Let \(u\) solve the initial value problem
\[
\begin{aligned}
  u_{tt}-u_{xx} &= 0, && \text{in }\R\times(0,\infty),\\
  u=g,\quad u_t=h, &&& \text{on }\R\times\{0\}.
\end{aligned}
\]
Suppose \(g,h\) have compact support. Define the kinetic energy
\(k(t)=\frac12\int_{-\infty}^{\infty}u_t^2(x,t)\dd x\) and the potential energy
\(p(t)=\frac12\int_{-\infty}^{\infty}u_x^2(x,t)\dd x\). Prove
\begin{enumerate}
\item[(a)] The total energy \(E=k(t)+p(t)\) is constant in \(t\).
\item[(b)] Assume that there are two distinct solutions \(u,v\) to the initial value
problem. By considering the energy of the difference solution \(U=u-v\), prove that
\(u=v\), that is, the solution is unique.
\end{enumerate}