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

TL;DR

This paper investigates a time-fractional Fisher-KPP equation incorporating memory effects for tumor growth, establishing well-posedness, stability, and proposing a numerical scheme validated through simulations showing unique tumor dynamics.

Contribution

It introduces a novel fractional diffusion model for tumor growth, proves well-posedness and stability, and develops a validated numerical method for simulation.

Findings

Global well-posedness for small initial data

Asymptotic stability of solutions

Numerical simulations reveal distinct tumor dynamics

Abstract

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.