Asymptotic behavior and representation of solutions to a Volterra kind of equation with a singular kernel

Abstract

Let $A$ be a densely defined closed, linear $\omega$-sectorial operator of angle $\theta \in [0,\frac{\pi }{2})$ on a Banach space $X$ for some $\omega \in \mathbb{R}$. 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: $u^{\prime }(t)=Au(t)+(\kappa \ast Au)(t),t>0$, $u(0)=u_0$, associated with the (possible) singular kernel $\kappa (t)=\alpha e^{-\beta t}\frac{t^{\mu -1}}{\Gamma (\mu )},t>0$, where $\alpha \in \mathbb{R}$, $\alpha \ne 0$, $\beta \ge 0$ and $0<\mu \le 1$.