  The observations show that less massive the galaxies are, the higher on
average is their specific star formation rate (SSFR = SFR/Ms, Ms is the stellar
mass). Such a trend, called the 'SSFR downsizing' (SSFR-DS) phenomenon, is seen
for local and high-z (back to z~1-2) galaxy samples. We use observational data
related only to disc galaxies and explore the average SSFR change with z for
different masses. For Ms in the range ~10^9.5-10^10.5 Msun, the SSFR increases
with (1+z) to a power that barely depends on Ms, and at all z's smaller
galaxies have ever higher SSFRs. The latter strongly disagree with the LCDM
hierarchical mass accretion rates. By means of self-consistent models of disc
galaxy evolution inside growing LCDM halos, the effects that disc
feedback-driven outflows and gas re-accretion have on the galaxy SSFR histories
are explored. The parameters of the outflow and re-accretion schemes are tuned
to reproduce the z~0 Mh-Ms relation inferred from observations. In the case of
outflows only, the SSFR of individual model galaxies is roughly proportional to
(1+z)^2.2 for all the masses with a normalization factor that depends on mass
as Ms^0.1, i.e more massive galaxies have slightly larger SSFRs, contrary to
the observed strong SSFR-DS trend. For the re-accretion cases, the dependence
on z remains approximately the same as without re-infall, but the correlation
on Ms even increases for most of the reasonable values of the model parameters.
The comparison of models and observations in the SSFR-Ms plane at z~0 shows the
divergent trend in SSFR as lower are the masses (upsizing vs downsizing). We
explain why the models show the reported trends, and conclude that the SSFR-DS
for low-mass galaxies poses a challenge for LCDM-based disc galaxy evolution
models. (Abridged)
