Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 105 / 3 / a

For a smooth function $g$ define its spherical meanThe Kirchhoff formuladefines a smooth solution of the three-dimensional wave equation for all positive and negative $t$ and has the prescribed data at $t=0$.

For uniqueness, apply the local energy estimate to the difference of two solutions on a backward light cone. Its energy at the cone tip is bounded by the zero initial energy on the cone base, so the difference and its derivatives vanish. Covering spacetime by such cones proves uniqueness among $C^2$ solutions.

Compact support is unnecessary for existence or uniqueness: the sphere in the Kirchhoff formula is compact for each $(t,x)$, so arbitrary smooth data suffice, and the cone-energy proof is local.