Asymptotic behavior of solutions of a time-space fractional diffusive Volterra equation

TL;DR

This paper investigates the long-term behavior of solutions to a complex time-space fractional Volterra differential equation, establishing boundedness, continuity, and asymptotic properties under specific boundary and initial conditions.

Contribution

It introduces an analysis of a novel fractional Volterra equation with Neumann boundary conditions, focusing on asymptotic behavior and solution boundedness.

Findings

Solutions are bounded and uniformly continuous over time.

Asymptotic behavior of positive solutions is characterized.

The study extends understanding of fractional differential equations with Volterra terms.

Abstract

In this paper, we study the time-space fractional differential equation of the Volterra type: \begin{align*} {D}^\alpha_{0 \vert t} (u) +(-\Delta_N)^{\sigma}u &= u(1+au-bu^2)-au\int_0^t {K}(t-s) u(\cdot) \, ds, \end{align*} where $a,b>0$ are given constants, $\alpha,\sigma \in (0,1)$, equipped with a homogeneous Neumann's boundary condition and a positive initial data. The boundedness and uniform continuity of the solution on the entire $\mathbb{R}^+$ are established. Moreover, the asymptotic behavior of the positive solution is investigated.