= Solution
For $t>0$, the three-dimensional <Kirchhoff formula> with zero displacement is
$$
\boxed{u(t,x)=\frac1{4\pi t}\int_{|y-x|=t}g(y)\,dS_y
=\frac t{4\pi}\int_{S^2}g(x+t\omega)\,d\omega}.
$$
The <Strong Huygens principle> says that the value depends only on the initial velocity on the sphere of radius $t$, not on points strictly inside it. In particular a compactly supported disturbance leaves no tail at a <fixed point> after the sphere has passed the support.
For <method of descent for the wave equation>, extend the two-dimensional initial velocity independently of the third coordinate. Projecting the three-dimensional sphere onto its planar disk gives two sheets, each with area element $t\,dy/\sqrt{t^2-|y-x|^2}$. Therefore
$$
\boxed{u(t,x)=\frac1{2\pi}\int_{|y-x|<t}
\frac{g(y)}{\sqrt{t^2-|y-x|^2}}\,dy}.
$$
The local argument can use <compact> cutoffs in the extra coordinate, so global Schwartz decay there is unnecessary. The <weak Huygens principle> is <finite propagation speed>: only data in the closed disk of radius $t$ can influence the point. Unlike the three-dimensional free equation, the full interior contributes and a tail can persist.
For the equation with potential $|x|^2$, multiply by $u_t$ and use the product rule:
$$
\partial_t\left[\tfrac12(u_t^2+|\nabla u|^2+|x|^2u^2)\right]
-\nabla\cdot(u_t\nabla u)=u_t(u_{tt}-\Delta u+|x|^2u)=0.
$$
This is the claimed local <energy> conservation with flux $p=-u_t\nabla u$.
Fix $(t_0,x_0)$ and let $E(t)$ be the integral of $e$ over the shrinking ball $|x-x_0|\le t_0-t$, $0\le t\le t_0-a$ with $a>0$. Its boundary moves inward at speed one. The <divergence theorem> gives
$$
E'(t)=\int_{\partial B}(u_t\partial_\nu u-e)\,dS\le0,
$$
because the negative integrand is
$$
-\tfrac12(u_t-\partial_\nu u)^2
-\tfrac12|\nabla_{\rm tan}u|^2-\tfrac12|x|^2u^2.
$$
All initial data are zero, so $E(0)=0$. Nonnegativity then forces $E(t)=0$. Letting the small terminal ball approach the cone tip proves $u=0$ there; alternatively $u_t=0$ throughout the cone and the initial value fixes $u$. The same argument applied to a difference of solutions whose data agree on the initial ball proves local domain of dependence. Thus \b[the potential equation has propagation speed at most one and satisfies the <weak Huygens principle>], without requiring finite total <energy> on all space. It does not assert the strong no-tail property.
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