From anomalous to classical diffusion in a non-linear heat equation

TL;DR

This paper investigates the transition from anomalous to classical diffusion in a nonlinear heat equation, proving uniform convergence of solutions and deriving convergence rates as the fractional parameter approaches 2.

Contribution

It provides a rigorous proof of solution convergence from fractional to classical Laplacian cases and quantifies the convergence rate.

Findings

Solutions of anomalous diffusion converge uniformly to classical diffusion solutions as alpha approaches 2.

Derived explicit convergence rate matching previous experimental observations.

Established mathematical framework for analyzing diffusion transition in nonlinear heat equations.

Abstract

In this paper, we consider the heat equation with the natural polynomial non-linear term; and with two different cases in the diffusion term. The first case (anomalous diffusion) concerns the fractional Laplacian operator with parameter $1<\alpha <2$, while, the second case (classical diffusion) involves the classical Laplacian operator. When $\alpha \to 2$, we prove the uniform convergence of the solutions of the anomalous diffusion case to a solution of the classical diffusion case. Moreover, we rigorous derive a convergence rate, which was experimentally exhibit in previous related works.