Effect of different additional $L^{m}$ regularity on semi-linear damped $\sigma$-evolution models

The motivation of the present study is to discuss the global (in time) existence of small data solutions to the following semi-linear structurally damped $\sigma$-evolution models: \begin{equation*} \partial_{tt}u+(-\Delta)^{\sigma}u+(-\Delta)^{\sigma/2}\partial_{t}u=\left|u\right| ^{p}, \ \sigma\geq 1, \ \ p>1, \end{equation*} where the Cauchy data $(u(0,x), \partial_{t}u(0,x))$ will be chosen from energy space on the base of $L^{q}$ with different additional $L^{m}$ regularity, namely \begin{equation*} u(0,x)\in H^{\sigma,q}(\mathbb{R}^{n})\cap L^{m_{1}}(\mathbb{R}^{n}) , \ \ \partial_{t}u(0,x)\in L^{q}(\mathbb{R}^{n})\cap L^{m_{2}}(\mathbb{R}^{n}), \ \ q\in(1,\infty),\ \ m_{1}, m_{2}\in [1,q). \end{equation*} Our new results will show that the critical exponent which guarantees the global (in time) existence is really affected by these different additional regularities and will take \textit{two different values} under some restrictions on $m_{1}, m_{2}$, $q$, $\sigma$ and the space dimension $n\geq1$. Moreover, in each case, we have no loss of decay estimates of the unique solution with respect to the corresponding linear models.