  We consider the following Cauchy problem for weakly coupled systems of
semi-linear damped elastic waves with a power source non-linearity in
three-dimensions: \begin{equation*}
  U_{tt}-a^2\Delta U-\big(b^2-a^2\big)\nabla\text{div }
U+(-\Delta)^{\theta}U_t=F(U),\,\, (t,x)\in[0,\infty)\times\mathbb{R}^3,
\end{equation*} where
$U=U(t,x)=\big(U^{(1)}(t,x),U^{(2)}(t,x),U^{(3)}(t,x)\big)^{\mathrm{T}}$ with
$b^2>a^2>0$ and $\theta\in[0,1]$. Our interests are some qualitative properties
of solutions to the corresponding linear model with vanishing right-hand side
and the influence of the value of $\theta$ on the exponents $p_1,p_2,p_3$ in
$F(U)=\big(|U^{(3)}|^{p_1},|U^{(1)}|^{p_2},|U^{(2)}|^{p_3}\big)^{\mathrm{T}}$
to get results for the global (in time) existence of small data solutions.
