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

We consider an integro-differential counterpart of the $\sigma -$evolution equation of the type $${\mathrm{\partial }}_t^{2}u(t,x)+\mu (-\mathrm{\Delta }{)}^{\frac{\sigma }{2}}{\mathrm{\partial }}_{t}u(t,x)+(-\mathrm{\Delta }{)}^{\sigma }u(t,x)=f(t,x),$$ with $\sigma >0$ and $\mu >0$, that encodes memory of $\mathit{\text{power-law}}$ type. To do so, we replace the time derivatives ${\mathrm{\partial }}_t$ and ${\mathrm{\partial }}_t^2$ by the so-called Caputo-Djrbashian derivatives ${\mathrm{\partial }}_t^{\gamma }$ of order $\gamma =\alpha$ and $\gamma =2\alpha$, respectively, and the inhomogeneous term $f(t,x)$ by the Riemann-Liouville integral $I_{0^{+}}^{\beta -2\alpha }f(t,x)$, whereby $0<\alpha \le 1$ and $2\alpha \le \beta <2\alpha +1$. 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 ${\displaystyle (-\mathrm{\Delta }{)}^{\frac{\eta }{2}}{E}_{\alpha ,\beta }(-\lambda (-\mathrm{\Delta }{)}^{\frac{\sigma }{2}}{t}^{\alpha })}$, endowed by the fractional Laplacian $-(-\mathrm{\Delta }{)}^{\frac{\sigma }{2}}$ 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.