Delayed Feedback and Asymptotic Decay for a Time-Fractional Equation with the Spectral Fractional Laplacian

We study a delayed semilinear evolution equation with a Caputo time derivative and the spectral fractional Dirichlet Laplacian on a bounded connected domain. The model separates two forms of memory: the Caputo operator retains the distributed Volterra history, whereas the nonlinear production samples the single past state u(t−τ). Working in the strongly continuous phase space C0(Ω), we prove local well-posedness, positivity, a sup-norm continuation criterion, and a compatible weak formulation. In the delayed-source case with μ=0, the solution exists globally and remains bounded on every finite time interval, while the first Dirichlet mode admits an explicit recursive sequence of positive lower bounds across successive delay windows. In the dissipative case μ>0, p>q>1, histories satisfying the explicit smallness conditions remain in an invariant order interval and the L2-energy decays at a Mittag–Leffler rate. The scalar computations are presented only as heuristic first-mode surrogate experiments. In addition, an independent spatially resolved sine spectral-Galerkin/L1 computation of the PDE, with temporal and spectral refinement studies, is included as a numerical illustration.