Lemma: Local zero-data wave problem

Let

For a smooth source on ${\mathcal{D}}_{R,b}$, the wave and Klein–Gordon equations with prescribed smooth initial data have unique smooth solutions on that domain. On every compact target they depend only on the data and source in its truncated past. The solution maps are continuous in the local smooth topology.

Proof. For compact spatial data and sources, spatial Fourier transformation gives the cosine and sine evolution, with

For example, zero-data solutions of $Pu=F$ are

The multiplier has a continuous massless value at zero frequency.

Finite propagation follows by integrating the usual wave energy on a truncated backward cone. For the homogeneous difference of two solutions, the lateral flux has the outward nonnegative sign, and zero initial energy gives zero energy throughout the cone. The same argument includes the nonnegative mass term.

For a compact target in ${\mathcal{D}}_{R,b}$, its initial shadow lies strictly inside $B_R$. Extend its required data and source smoothly with compact cutoffs, solve the compact problem, and restrict. Two choices agree by the cone uniqueness just proved. An exhaustion by compact targets gives a compatible smooth solution on the full domain. Differentiated energy estimates and Sobolev estimates on compact cones give continuous dependence in every smooth seminorm.