• Physically consistent time-fractional Fisher-KPP equation is analyzed. • Local weak well-posedness is proved by Galerkin compactness methods. • Global existence is obtained for sufficiently small initial data. • A graded convolution-quadrature FEM is developed for the model. • Simulations distinguish the model from the Caputo-in-time formulation. We study a time-fractional Fisher–KPP equation involving a Riemann–Liouville fractional derivative acting on the diffusion term, as derived by Angstmann and Henry (Entropy, 22:1035, 2020). The model captures memory effects in diffusive population dynamics and serves as a framework for tumor growth modeling. We first establish local well-posedness of weak solutions. The analysis combines a Galerkin approximation with a refined a priori estimate based on a Bihari–Henry–Gronwall inequality, addressing the nonlinear coupling between the fractional diffusion and the reaction term. For small initial data, we further prove global well-posedness and asymptotic stability. A numerical method based on a nonuniform convolution quadrature scheme is then proposed and validated. Simulations demonstrate distinct dynamical behaviors compared to conventional formulations, emphasizing the physical consistency of the present model in describing tumor progression.
Fritz et al. (2026) studied this question.