My Problem is about the following question found at Bounding the solution of a wave equation in 3 dimensions: : Let $u: \mathbb{R}^3 \times (0,\infty) \rightarrow \mathbb{R}$ be a solution of the Cauchy Problem:
$u_{tt} - \Delta u = 0 (x,t) \in \mathbb{R}^3 \times (0,\infty); \\ u(x,0) = g; u_t(x,0) = h$ with $g,h \in C^{\infty}_0(\mathbb{R}^3)$ (g,h smooth with compact support)
Show that there exists a $C>0$ such that $|u(x,t)| <= C/t; (x,t) \in \mathbb{R}^3 \times (0,\infty)$
So the answer given at the link above (https://math.stackexchange.com/a/742177/1170005) says that if you use Kirchoff's formula:
"Since $g,h,\nabla g$ are compactly supported, there exists $M>0$ such that $g,h, \nabla g=0$ on $S(x,t)$ for all $t⩾M$." with $S(x,t)$ denoting the boundary of the ball with radius $t$ and center $x$.
My Question is: I think $M$ in the answer given at the link above is dependend on $x$, because for different $x$ $M$ might vary, and thus the constant $C$ derived in the answer above is also dependend on $x$. Is this correct or did i make a mistake?
Thank you for your help :)
Source: Evans, partial differential equations, problem 18 on page 8