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

We study a delayed time-fractional semilinear evolution equation driven by the spectral fractional Laplacian. The model combines a Caputo time derivative, a source evaluated at the past state u(t−τ), and an absorption term evaluated at the present state u(t). The fractional structure is used throughout the analysis through the Caputo Volterra kernel, the spectral fractional energy form, and Mittag–Leffler resolvent estimates. We prove local existence, positivity and an L∞(Ω) continuation criterion for bounded mild solutions, and show that these solutions satisfy the weak formulation before the maximal time. In the pure delayed-source case μ=0, we prove global continuation and boundedness on every finite time interval, together with a stepwise propagation of positive lower bounds for the first Dirichlet mode. When μ>0 and the initial history is sufficiently small, the present absorption and the spectral damping dominate the delayed feedback. In this regime, the solution is globally bounded and satisfies u(t)L2(Ω)→0 as t→+∞, with a Mittag–Leffler type decay estimate for the L2-energy. The numerical section is restricted to reduced first-mode comparison computations and illustrates the scalar comparison estimates derived from the first-mode analysis.