Asymptotic behavior of solutions of a time-space fractional diffusive Volterra equation
Sofwah Ahmad, Mokhtar Kirane
TL;DR
We analyze a time-space fractional Volterra equation with Caputo time derivative $D^\alpha_{0|t}$ and spectral fractional Laplacian $(-\Delta_N)^\sigma$ under Neumann boundary conditions on a bounded domain, including a nonlocal delay term with kernel $\mathcal{K}$. The study develops a mild-solution framework via $S_\alpha(t)$ and $P_\alpha(t)$, proves local existence and positivity, and establishes global boundedness and uniform continuity in time. Using Langlais-Phillips dynamics and precompact orbits, the authors show that every positive solution converges to the positive equilibrium $u_\infty = 1/\sqrt{b}$, as the long-time limit of the associated elliptic problem $(-\Delta_N)^\sigma \varphi = \varphi(1-b\varphi^2)$. A spectral argument around the zero equilibrium confirms instability there, reinforcing the global attractivity toward the positive constant state. These results illuminate the asymptotic behavior of nonlocal-in-time and nonlocal-in-space population models with fractional dynamics.
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, $α,σ\in (0,1)$, 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.