  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.
