Spose $ u $ solves the wave equation on $ U \subset \mathbb{R}^3 $ with initial conditions $ u (x, 0) = g(x)$ and $ u_t(x,0) = h(x)$, where lower script indicates partial differentiation. Then we have
$\int _U u_t^2 + |\nabla u|^2 \leq \int _U h^2 + |\nabla g|^2 $.
My question is then does this give me some bounds (control) on $u$ in terms of $g$ and $h$? If so how do I show this?
Thanks!