If $u$ is a solution to the wave equation (Cauchy) then $|u(x,t)|\le A/t$ for some $A$

Question

Let $u(x,t)$ be a solution for the Cauchy Problem

$$u_{tt}-\Delta_xu = 0\mbox{ in $\mathbb{R}^3\times \mathbb{R}$}$$ $$u(x,0) = f(x)\mbox{ in $\mathbb{R}^3$}$$ $$u_t(x,0) = g(x) \mbox{in $\mathbb{R}^3$}$$

where $f$ and $g$ are of class $C^3$ and $C^2$ respectively in $\mathbb{R}^3$ which are null in the complementar of a compact. Show that there exists a constant $A$ such that

$$|u(x,t)|\le A/t, x\in\mathbb{R}^3, t\ge 1$$

Find, also, an estimative for the constant $A$ in terms of $f$ and $g$.

UPDATE:

I've found the solution

but I need to understand why the intersection with the support is at most $4\pi R^2$. As I understand, the support of the data is the support of $f$. The intersection of $S_t(x)$ with this support should be what?

I'm trying to imagine the complement of the ball $B(0,R)$ which contains the support of $f$ (is the support $g$ necessary?). I must take the intersection with $S_x(t)$ but I do not know what to do

Answer 1

We use the following fact.

• If $C\subset\mathbb R^3$ is a convex set contained in a ball $B_R$, then $\mathcal H^2(\partial C)\leq \mathcal H^2(\partial B_R)=4\pi R^2$.

With your notation, let $E_x(t)$ be the ball bounded by $S_x(t)$ and let $B_R$ the ball that contains $\mathrm{supp}(f)$ and $\mathrm{supp}(g)$. Then $E_x(t)\cap B_R$ is convex and contained in $B_R$. Its surface is composed of two pieces, one of which is $\partial E_x(t)\cap B_R = S_x(t)\cap B_R$. Therefore $$ \mathcal H^2\bigl(S_x(t)\cap B_R\bigr) \leq \mathcal H^2\bigl(\partial(E_x(t)\cap B_R)\bigr) \leq \mathcal H^2(\partial B_R)=4\pi R^2 . $$

Edit

Obviously, I applied the initial fact with $C=E_x(t)\cap B_R$.