Reaction-Diffusion System Approximation to the Fast Diffusion Equation

TL;DR

This paper introduces a reaction-diffusion system approximation for the fast diffusion equation, establishing its mathematical properties and validating its effectiveness through numerical experiments.

Contribution

It develops a novel semilinear reaction-diffusion approximation for singular diffusion problems and proves its convergence to the true solution under multiple regimes.

Findings

The approximation system is well-posed and admits uniform a priori estimates.

Convergence to the weak solution is established in three asymptotic regimes.

Numerical experiments confirm the theoretical convergence and practical utility.

Abstract

This paper proposes a novel reaction-diffusion system approximation tailored for singular diffusion problems, typified by the fast diffusion equation. While such approximation methods have been successfully applied to degenerate parabolic equations, their extension to singular diffusion-where the diffusion coefficient diverges at low densities-has remained unexplored. To address this, we construct an approximating semilinear system characterized by a reaction relaxation parameter and a time-derivative regularizing parameter. We rigorously establish the well-posedness of this system and derive uniform a priori estimates. Using compactness arguments, we prove the convergence of the approximate solutions to the unique weak solution of the target singular diffusion equation under three distinct asymptotic regimes: the simultaneous limit, the limit via a parabolic-elliptic system, and the limit via a uniformly parabolic equation. This approach effectively transfers the diffusion singularity into the reaction terms, yielding a highly tractable system for both theoretical analysis and computation. Finally, we present numerical experiments that validate our theoretical convergence results and demonstrate the practical efficacy of the proposed approximation scheme.