Let A be a densely defined closed, linear $$\omega$$ ω -sectorial operator of angle $$\theta \in [0,\frac{\pi }{2})$$ θ ∈ [ 0 , π 2 ) on a Banach space X , for some $$\omega \in \mathbb {R}$$ ω ∈ R . We give an explicit representation (in terms of some special functions) and study the precise asymptotic behavior as time goes to infinity of solutions to the following diffusion equation with memory: $$\displaystyle u'(t)=Au(t)+(\kappa *Au)(t), \, t >0$$ u ′ ( t ) = A u ( t ) + ( κ ∗ A u ) ( t ) , t > 0 , $$u(0)=u_0$$ u ( 0 ) = u 0 , associated with the (possible) singular kernel $$\kappa (t)=\alpha e^{-\beta t}\frac{t^{\mu -1}}{\Gamma (\mu )},\;\;t>0$$ κ ( t ) = α e - β t t μ - 1 Γ ( μ ) , t > 0 , where $$\alpha \in \mathbb {R}$$ α ∈ R , $$\alpha \ne 0$$ α ≠ 0 , $$\beta \ge 0$$ β ≥ 0 and $$0<\mu < 1$$ 0 < μ < 1 .

中文翻译：

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具有奇异核的 Volterra 类方程的渐近行为和解的表示

设 A 是一个密集定义的闭合线性 $$\omega$$ ω -角 $$\theta \in [0,\frac{\pi }{2})$$ θ ∈ [ 0 , π 2 ) 的扇区算子在 Banach 空间 X 上，对于一些 $$\omega \in \mathbb {R}$$ ω ∈ R 。我们给出了一个明确的表示（就一些特殊函数而言），并研究了随着时间的推移，具有记忆的以下扩散方程的无限解的精确渐近行为：$$\displaystyle u'(t)=Au(t)+( \kappa *Au)(t), \, t >0$$ u ′ ( t ) = A u ( t ) + ( κ ∗ A u ) ( t ) , t > 0 , $$u(0)=u_0 $$ u ( 0 ) = u 0 ，与（可能的）奇异核 $$\kappa (t)=\alpha e^{-\beta t}\frac{t^{\mu -1}}{\ Gamma (\mu )},\;\;t>0$$ κ ( t ) = α e - β tt μ - 1 Γ ( μ ) , t > 0 , 其中 $$\alpha \in \mathbb {R} $$ α ∈ R , $$\alpha \ne 0$$ α ≠ 0 , $$\beta \ge 0$$ β ≥ 0 和 $$0<\mu < 1$$ 0 < μ < 1 。