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

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

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

$$|u(x,t)|le A/t, xinmathbb{R}^3, tge 1$$

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 $4pi R^2$. 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

We use the following fact.

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

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
$$
mathcal H^2bigl(S_x(t)cap B_Rbigr)
leq mathcal H^2bigl(partial(E_x(t)cap B_R)bigr)
leq mathcal H^2(partial B_R)=4pi R^2 .
$$

Edit

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