Lemma: Supported smooth Cauchy evolution

For $\nu \ge 0$, smooth data supported in ${\overline{B}}_{r_{\ast }}$, and smooth forcing supported in ${\mathcal{C}}_{r_{\ast }}$, the equation

has a unique smooth solution on $\hat{\mathcal{L}}$. It is supported in ${\mathcal{C}}_{r_{\ast }}$. Evolution is continuous in the compact-data smooth topology and the compact-open smooth topology on each closed subinterval of $I$.

If $F=0$ for $s\le 0$, its zero-data solution is zero there.

Proof. Let ${\Omega }_{\nu }=(-\Delta +{\nu }^2{)}^{1/2}$. Spatial Fourier transformation gives

The sine quotient has its continuous value at zero frequency. On compact time intervals the source has uniformly compact spatial support, so the formula and its differentiated versions are well defined in Sobolev spaces of every order.

For integer $n\ge 0$, an energy containing the $L^2$ norms of ${\partial }_s{\partial }^{\beta }u$, $\nabla {\partial }^{\beta }u$, and ${\partial }^{\beta }u$, $|\beta |\le n$, satisfies

in the forward direction, with the corresponding backward estimate. Gronwall's inequality gives uniqueness and continuous dependence. Sobolev embedding and the equation give smooth control.

The local energy identity outside the expanding initial ball has nonnegative outward characteristic flux. The source bound $|x|\le r_{\ast }+|\sigma |$ therefore keeps the solution in ${\mathcal{C}}_{r_{\ast }}$, in either time direction. Zero forcing and zero data on the past side give zero there.