I am interested in the wave equation for $\mathbb{R}^3 $\begin{cases}u_{tt}-u_{xx}=0 \\ u(x,0)=\phi(x) \\u_t(x,0)=\psi(x)\end{cases}
with $\phi,\psi\in C^1(\mathbb{R}^3)$
Using Kirchoff's formula, the solution is:
$u(x,t)=(4\pi t^2)^{-1}\int_{\partial B(x,t)}[\phi(y)+\nabla \phi(y)(y-x)+t\psi(y)]dS(y)$. I'm trying to show the estimate of the form:
$\int_{0}^{\infty}u^2(0,t)dt\leq (4\pi)^{-1}\int_{\mathbb{R^3}}(u_t(x,0))^2 dx$.
I tried the simple case where $\phi=0$ but even I am unsure how to do it for the simple case. I know it can be shown $|u(x,t)|\leq \frac{K}{t}$, which means $|u(x,t)|^2=u^2(x,t)\leq \frac{K^2}{t^2}$, in particular $u^2(0,t)\leq \frac{K^2}{t^2}$, and here is where I get stuck.