Structurally damped $\sigma-$evolution equations with power-law memory

We consider an integro-differential counterpart of the $\sigma-$evolution equation of the type $$\partial_t^2 u(t,x)+\mu (-\Delta)^{\frac{\sigma}{2}} \partial_t u(t,x)+(-\Delta)^\sigma u(t,x)=f(t,x),$$ with $\sigma>0$ and $\mu>0$, that encodes memory of \textit{power-law} type. To do so, we replace the time derivatives $\partial_t$ and $\partial_t^2$ by the so-called Caputo-Djrbashian derivatives $\partial_t^\gamma$ of order $\gamma=\alpha$ and $\gamma=2\alpha$, respectively, and the inhomogeneous term $f(t,x)$ by the Riemann-Liouville integral $I^{\beta-2\alpha}_{0^+}f(t,x)$, whereby $0<\alpha\leq 1$ and $2\alpha\leq \beta<2\alpha+1$. For the solution representation of the underlying Cauchy problems on the space-time $[0,T]\times \mathbb{R}^n$ we then consider a wide class of pseudo-differential operators $\displaystyle (-\Delta)^{\frac{\eta}{2}}E_{\alpha,\beta}\left(~-\lambda(-\Delta)^{\frac{\sigma}{2}} t^\alpha~\right)$, endowed by the fractional Laplacian $-(-\Delta)^{\frac{\sigma}{2}}$ and the two-parameter Mittag-Leffler functions $E_{\alpha,\beta}$. On our approach we are also able to provide dispersive and Strichartz estimates for the solutions with the aid of decay properties of $E_{\alpha,\beta}(-z)$ ($z\in \mathbb{C}$) and the boundedness properties of the Hankel transform.