Pinning and de-pinning phenomena in front propagation in heterogeneous media

Abstract

This paper investigates the pinning and de-pinning phenomena of some evolutionary partial differential equations which arise in the modelling of the propagation of phase boundaries in materials under the combined effects of an external driving force $F$ and an underlying heterogeneous environment. The phenomenology is the existence of pinning states – stationary solutions – for small values of $F,$ and the appearance of genuine motion when $F$ is above some threshold value. In the case of a periodic medium, we characterise quantitatively, near the transition regime, the scaling behaviour of the interface velocity as a function of $F$. The results are proved for a class of some semi-linear and reaction-diffusion equations.