we know $u$ (if it is a solution to the wave equation in $\mathbb{R}^3$) decays as $1/t$ as $t$ goes to $\infty$. this comes easily from spherical means. but how do we know this is the maximum possible rate of decay?



maximum decay rate
Let $\bar u(t,r)$ be the spherical mean over the sphere of radius $r$, centered at (say) the origin. Then $r\bar u(t,r)$ satisfies the one-dimensional wave equation. If $u$ has compact support initially, then eventually $r\bar u(t,r)$ is an outward traveling wave. This wave has constant shape, and so $\bar u(t,r)$ decays as $1/r$.


