A time-fractional Fisher–KPP equation for tumor growth: Analysis and numerical simulation

• 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.