can you guys give me an idea on how to begin solving this problem?
suppose $u(x,t)$ is smooth and solves
$u_{tt}(x,t)-4\Delta_{x}u(x,t)=0$, $x\in \mathbb{R}^{3}, t>0$
compute the radial solutions $u(x,t)=v(r,t), r=\sqrt{x_{1}^{2}+x_{1}^{2}+x_{3}^{2}}$
Also, is $\Delta_{x}$ a typo? should it say just $\Delta$?
In spherical coordinates, the Laplacian for a radially symmetric function $f=f(r)$ takes the form:
$$\Delta f = \frac{1}{r}\frac{\partial^2}{\partial r^2}(rf).$$
So, if you search for a solution $v=v(r,t)$ then $$ v_{tt} - 4\Delta v = v_{tt} - \frac{4}{r}\frac{\partial^2}{\partial r^2}(rv)=0.$$
Multiplying by $r$, and noting that $rv_{tt} = (rv)_{tt}$, we have
$$ \frac{\partial^2}{\partial t^2}(rv) - 4 \frac{\partial^2}{\partial r^2}(rv)=0.$$
Changing variables: $u = rv$ it follows $$ u_{tt} - 4u_{rr} =0, $$ so that $u$ solves the 1D wave equation. General solutions to the 1D wave equation are
$u(r,t) = F(r-2t)+ G(r+2t),$
so we must have $$ v(r,t) = \frac{1}{r}(F(r-2t)+G(r+2t)), $$ where $F$ and $G$ are arbitrary.