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

TL;DR

This paper analyzes the asymptotic behavior of solutions to a diffusion equation with memory involving a singular kernel, providing explicit representations and detailed long-term behavior on Banach spaces.

Contribution

It offers an explicit solution representation and detailed asymptotic analysis for a diffusion equation with a singular kernel and memory effects.

Findings

Explicit solution representation using special functions

Precise asymptotic behavior as time approaches infinity

Analysis applicable to operators with sectorial properties

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: $\displaystyle u'(t)=Au(t)+(\kappa\ast Au)(t), \, t >0$, $u(0)=u_0$, associated with the (possible) singular kernel $\kappa(t)=\alpha e^{-\beta t}\frac{t^{\mu-1}}{\Gamma(\mu)},\;\;t>0$, where $\alpha\in\R$, $\alpha\ne 0$, $\beta\ge 0$ and $0<\mu\le 1$.