  Recent evidence based independently on spectral line strengths and dynamical
modelling point towards a non-universal stellar Initial Mass Function (IMF),
probably implying an excess of low-mass stars in elliptical galaxies with a
high velocity dispersion. Here we show that a time-independent bottom-heavy IMF
is compatible neither with the observed metal-rich populations found in giant
ellipticals nor with the number of stellar remnants observed within these
systems. We suggest a two-stage formation scenario involving a time-dependent
IMF to reconcile these observational constraints. In this model, an early
strong star-bursting stage with a top-heavy IMF is followed by a more prolonged
stage with a bottom-heavy IMF. Such model is physically motivated by the fact
that a sustained high star formation will bring the interstellar medium to a
state of pressure, temperature and turbulence that can drastically alter the
fragmentation of the gaseous component into small clumps, promoting the
formation of low-mass stars. This toy model is in good agreement with the
different observational constrains on massive elliptical galaxies, such as age,
metallicity, alpha-enhancement, M/L, or the mass fraction of the stellar
component in low-mass stars.
