An increasing amount of recent observational evidence shows that the less massive galaxies are, the higher on average their specific star-formation rates (SSFR = SFR/M-*, where M-* is the stellar mass). Such a trend, called the 'SSFR downsizing' (SSFR-DS) phenomenon, is seen for local and high-redshift (back to z similar to 1-2) galaxy samples. We use observational data related only to disc galaxies and explore how the average SSFR changes with z for different masses. For all masses in the range similar to 109.5-1010.5 M-circle dot, the SSFR increases with (1 + z) to a power that seems not to depend on M-*, and at all redshifts smaller galaxies always have higher SSFRs; galaxies less massive than M-* similar to 1010 M-circle dot are now forming stars at a greater rate than in the past, assuming constant SFRs over a Hubble time to build stellar mass. We show that these features strongly disagree with the Lambda Cold Dark Matter (Lambda CDM) halo hierarchical mass accretion rates. Further, by means of self-consistent models of disc galaxy evolution inside growing Lambda CDM haloes, the effects of disc-feedback-driven outflows and gas re-accretion on galaxy SSFR histories are explored. The parameters of the outflow and re-accretion schemes are tuned to reproduce the present-day M-h-M-* relation (where M-h is the halo mass) inferred from the observationally based M-* function of disc galaxies. In the case of outflows only, the SSFR of individual model galaxies increases with z roughly as (1 + z)2.2 for all masses (somewhat shallower than observations) with a normalization factor that depends on mass as M0.1(*), 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 gas re-infall, but the correlation with mass increases even for the most reasonable values of the model parameters. The comparison of models and observations in the SSFR-M-* plane at z similar to 0 (where the data are more reliable) clearly shows the divergent trend in SSFR when the masses are lower (upsizing versus downsizing). We explain why our models show the reported trends, and conclude that the SSFR-DS phenomenon for low-mass galaxies poses a sharp challenge for Lambda CDM-based disc galaxy evolution models.
