Structurally damped $σ-$evolution equations with power-law memory

Abstract

We consider an integro-differential counterpart of the $σ-$evolution equation of the type \[ \partial_t^2 u(t,x)+μ(-Δ)^{\fracσ{2}} \partial_t u(t,x)+(-Δ)^σu(t,x)=f(t,x), \] with $σ>0$ and $μ>0$, 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 $\partial_t^γ$ of order $γ=α$ and $γ=2α$, respectively, and the inhomogeneous term $f(t,x)$ by the Riemann-Liouville integral $I^{β-2α}_{0^+}f(t,x)$, whereby $0<α\leq 1$ and $2α\leq β<2α+1$. 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 $\displaystyle (-Δ)^{\fracη{2}}E_{α,β}\left(~-λ(-Δ)^{\fracσ{2}} t^α~\right)$, endowed by the fractional Laplacian $-(-Δ)^{\fracσ{2}}$ and the two-parameter Mittag-Leffler functions $E_{α,β}$. On our approach we are also able to provide dispersive and Strichartz estimates for the solutions with the aid of decay properties of $E_{α,β}(-z)$ ($z\in \mathbb{C}$) and the boundedness properties of the Hankel transform.