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: \(\displaystyle u'(t)=Au(t)+(\kappa *Au)(t), \, t >0\), \(u(0)=u_0\), 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 \(0<\mu < 1\).