An extension can be constructed without renaming it

Let ${\mathcal{H}}_T$ be the real vector space of ambient smooth sources $q(t,\mathbf{x})=q(t)$, independent of space and zero for $t\le 0$, restricted to the slab. Regard ${\mathcal{D}}_{\mathrm{int}}$ as an ambient source by its smooth zero extension. Then

Indeed, a nonzero spatially constant function cannot be compactly supported in space.