Lemma: Smooth forced evolution

For $\mu \ge 0$, the equation

has a unique global smooth solution for smooth compact Cauchy data and smooth forcing with bounded spatial support on every finite time interval. On such intervals, the solution depends continuously on data and forcing in their smooth topologies. Finite propagation gives the corresponding cone-support bounds when the data and source satisfy those bounds.

Proof. Let ${\Omega }_{\mu }=\sqrt{-\Delta +{\mu }^2}$. Spatial Fourier transformation gives

For $\mu =0$, the sine quotient has the continuous value $t-s$ at zero frequency. On a finite interval, the source has uniformly compact spatial support and bounded smooth seminorms, so the required Sobolev integrals are defined. Differentiation verifies the equation and data.

For an integer $s\ge 0$, use

Integration by parts and Cauchy–Schwarz give

The backwards estimate and Gronwall's inequality give uniqueness and continuous dependence. Increasing $s$, applying Sobolev embedding, and using the equation for time derivatives give smooth estimates.

For finite propagation, integrate the local energy identity outside an expanding ball. The outward characteristic flux has the required nonnegative sign. Where initial data and forcing vanish, the exterior energy remains zero. The same argument after a time translation proves the retarded support statement. These are the flat-space estimates underlying the standard general Cauchy theory BGP (Chapter 3).