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

Abstract

In this paper, we study the time-space fractional differential equation of the Volterra type: \begin{align*} {D}^α_{0 \vert t} (u) +(-Δ_N)^σu &= 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.