  We study the following Cauchy problems for semi-linear structurally damped
$\sigma$-evolution models: \begin{equation*} u_{tt}+ (-\Delta)^\sigma u+ \mu
(-\Delta)^\delta u_t = f(u,u_t),\, u(0,x)= u_0(x),\, u_t(0,x)=u_1(x)
\end{equation*} with $\sigma \ge 1$, $\mu>0$ and $\delta \in
(0,\frac{\sigma}{2})$. Here the function $f(u,u_t)$ stands for the power
nonlinearities $|u|^{p}$ and $|u_t|^{p}$ with a given number $p>1$. We are
interested in investigating $L^{1}$ estimates for oscillating integrals in the
presentation of the solutions to the corresponding linear models with vanishing
right-hand sides by applying the theory of modified Bessel functions and
Fa\`{a} di Bruno's formula. By assuming additional $L^{m}$ regularity on the
initial data, we use $(L^{m}\cap L^{q})- L^{q}$ and $L^{q}- L^{q}$ estimates
with $q\in (1,\infty)$ and $m\in [1,q)$, to prove the global (in time)
existence of small data Sobolev solutions to the above semi-linear models from
suitable function spaces basing on $L^q$ spaces.
