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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 powernonlinearities $|u|^{p}$ and $|u_t|^{p}$ with a given number $p>1$. We areinterested in investigating $L^{1}$ estimates for oscillating integrals in thepresentation of the solutions to the corresponding linear models with vanishingright-hand sides by applying the theory of modified Bessel functions andFa\`{a} di Bruno's formula. By assuming additional $L^{m}$ regularity on theinitial data, we use $(L^{m}\cap L^{q})- L^{q}$ and $L^{q}- L^{q}$ estimateswith $q\in (1,\infty)$ and $m\in [1,q)$, to prove the global (in time)existence of small data Sobolev solutions to the above semi-linear models fromsuitable function spaces basing on $L^q$ spaces.

An application of $L^1$ estimates for oscillating integrals to parabolic like semi-linear structurally damped $\sigma$-evolution models