  We consider front solutions of the Swift-Hohenberg equation $\partial_t u=
-(1+\partial_x^2)^2 u +\epsilon ^2 u -u^3$. These are traveling waves which
leave in their wake a periodic pattern in the laboratory frame. Using
renormalization techniques and a decomposition into Bloch waves, we show the
non-linear stability of these solutions. It turns out that this problem is
closely related to the question of stability of the trivial solution for the
model problem $\partial_t u(x,t) = \partial_x^2 u
(x,t)+(1+\tanh(x-ct))u(x,t)+u(x,t)^p$ with $p>3$. In particular, we show that
the instability of the perturbation ahead of the front is entirely compensated
by a diffusive stabilization which sets in once the perturbation has hit the
bulk behind the front.
