  We consider asymptotically stable scalar-tensor dark energy (DE) models for
which the equation of state parameter $w_{DE}$ tends to zero in the past. The
viable models are of the phantom type today, however this phantomness is milder
than in General Relativity if we take into account the varying gravitational
constant when dealing with the SNIa data. We study further the growth of matter
perturbations and we find a scaling behaviour on large redshifts which could
provide an important constraint. In particular the growth of matter
perturbations on large redshifts in our scalar-tensor models is close to the
standard behaviour $\delta_m \propto a$, while it is substantially different
for the best-fit model in General Relativity for the same parametrization of
the background expansion. As for the growth of matter perturbations on small
redshifts, we show that in these models the parameter $\gamma'_0\equiv
\gamma'(z=0)$ can take absolute values much larger than in models inside
General Relativity. Assuming a constant $\gamma$ when $\gamma'_0$ is large
would lead to a poor fit of the growth function $f$. This provides another
characteristic discriminative signature for these models.
