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We consider an integro-differential counterpart of the $\sigma-$evolutionequation 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 replacethe time derivatives $\partial_t$ and $\partial_t^2$ by the so-calledCaputo-Djrbashian derivatives $\partial_t^\gamma$ of order $\gamma=\alpha$ and$\gamma=2\alpha$, respectively, and the inhomogeneous term $f(t,x)$ by theRiemann-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 solutionrepresentation 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 andStrichartz estimates for the solutions with the aid of decay properties of$E_{\alpha,\beta}(-z)$ ($z\in \mathbb{C}$) and the boundedness properties ofthe Hankel transform.

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