论文标题
结构阻尼$σ-$ Evolution方程，带有幂律存储器
Structurally damped $σ-$evolution equations with power-law memory
论文作者
论文摘要
我们考虑类型\ [\ partial_t^2 u（t，x）+μ（-Δ）^{\fracσ{2}} \ partial_t u（p partial_t u（t，x） $μ> 0 $，该编码\ textit {powerlaw}类型的内存。为此，我们用所谓的caputo-djrbashian衍生物$ \ partial_t^γ$替换了时间衍生物$ \ partial_t $和$ \ partial_t^2 $，分别是$γ=α$和$γ=α$和$γ=2α$的caputo-djrbashian衍生物$ \ partial_t^γ$ $ i^{β-2α} _ {0^+} f（t，x）$，其中$ 0 <α\ leq 1 $和$2α\ leqβ<2α+1 $。对于时空$ [0，t] \ times \ mathbb {r}^n $的解决方案表示的解决方案表示形式，我们然后考虑一类广泛的伪差异操作员$ \ displayStyle （-Δ）^{\fracη{2}} e_ {α，β} \ left（〜-λ（-Δ）^{\fracσ{2}} t^α〜 \ right）$，由分数laplacian laplacian $ - （ - δ）$ - （ - Δ）函数$ e_ {α，β} $。在我们的方法上，我们还能够借助$ e_ {α，β}（ - z）$（$ z \ in \ mathbb {c} $）和汉克尔变换的界限。
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.
