The destruction of anomalous diffusion of the Harper model at criticality, due to weak non-linearity chi is analyzed. It is shown that the second moment grows sub-diffusively as its expectation value is proportional to t to the power alpha, up to times t star proportional to chi to the power gamma. The exponents alpha and gamma reflect the multifractal properties of the spectra and eigenfunctions of the linear model. For t larger than t star the anomalous diffusion law is recovered, however the evolving profile has different shape with respect to the linear case. Applications to waveguide structures, and arrays of magnetic micro-traps for atomic Bose-Einstein condensates are discussed.