 Solution

Use unit speed and write the Cauchy data as $u(0)=f$, $u_t(0)=g$. For finite-energy data define the wave energy estimate quantityMultiply $u_{tt}-{\Delta }_{x}u=0$ by $u_t$ and use integration by parts. With compact support or sufficient decay the boundary flux is zero, soFor general finite-energy solutions, cutoff or approximation arguments justify this identity; equivalently the local estimate below, applied in both time directions and with radii tending to infinity, gives the same equality. Arbitrary smooth data need not have finite global energy; then the global bound with an infinite right side is uninformative, while the local estimate remains useful.

If $f\in H^1$ and $g\in L^2$, the fundamental theorem of calculus and the wave energy estimate further giveTogether these yield an a priori bound for $\Vert u(t){\Vert }_{H^1}+\Vert u_t(t){\Vert }_2$ on each bounded time interval. No existence assumption is proved by the estimate itself; it controls any sufficiently regular solution.