  This article investigates the long-time behaviour of parabolic scalar
conservation laws of the type $\partial_t u + \mathrm{div}_yA(y,u) - \Delta_y
u=0$, where $y\in\mathbb R^N$ and the flux $A$ is periodic in $y$. More
specifically, we consider the case when the initial data is an $L^1$
disturbance of a stationary periodic solution. We show, under polynomial growth
assumptions on the flux, that the difference between $u$ and the stationary
solution vanishes for large times in $L^1$ norm. The proof uses a self-similar
change of variables which is well-suited for the analysis of the long time
behaviour of parabolic equations. Then, convergence in self-similar variables
follows from arguments from dynamical systems theory. One crucial point is to
obtain compactness in $L^1$ on the family of rescaled solutions; this is
achieved by deriving uniform bounds in weighted $L^2$ spaces.
