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

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.