Sharp Spectral Stability Thresholds for Time–Space Fractional Reaction–Diffusion Equations with Hereditary Memory

We investigate the spectral stability of a class of time–space fractional reaction–diffusion equations with an additional hereditary convolution memory. The temporal dynamics are described by a Caputo derivative of order [Formula: see text], while spatial diffusion is generated by the spectral fractional Dirichlet Laplacian of order [Formula: see text]. For [Formula: see text] and nonnegative memory kernels [Formula: see text], a modal decomposition is used to derive an exact spectral stability criterion. We prove that the zero solution is spectrally stable if and only if [Formula: see text], where [Formula: see text] denotes the total hereditary mass. Equality corresponds to a critical zero characteristic root, whereas violation of the inequality produces a unique positive real root in the first spatial mode. For exponential memory, the instability transition occurs through a zero-frequency crossing. Although the stability threshold is independent of the temporal fractional order [Formula: see text], the unstable branch obeys a near-critical power law with exponent [Formula: see text]. Numerical computations verify the sharp threshold, the fractional onset law, and the fact that different nonnegative kernels with equal total mass share the same stability boundary while exhibiting different spectral branches away from criticality.