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For $t>0$, the three-dimensional Kirchhoff formula with zero displacement isThe 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 $tdy/\sqrt{t^2-|y-x{|}^2}$. ThereforeThe 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: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 givesbecause the negative integrand isAll 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 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.

The weak Huygens principle is the domain-of-dependence statement that initial data outside a backward light cone cannot influence its tip. It permits an interior tail and is weaker than the Strong Huygens principle. For $u_{tt}-\Delta u+V(x)u=0$ with $V\ge 0$, integrating the local energy over shrinking balls proves this statement: the boundary contribution is $u_t{\partial }_{\nu }u-\frac{1}{2}(u_t^2+|\nabla u{|}^2+V{u}^2)\le 0$.