One of the key unanswered questions in the study of galaxy evolution is what physical processes inside galaxies drive the changes in the SFRs in individual galaxies that, taken together, produce the large decline in the global star-formation rate density (SFRD) to redshifts since z~2. Many studies of the SFR at intermediate redshifts have been made as a function of the integrated stellar mass of galaxies but these did not use information on the internal structural properties of the galaxies. In this paper we present a comparative study of the dependence of SFRs on the average surface mass densities (SigmaM) of galaxies of different morphological types up to z~1 using the zCOSMOS and SDSS surveys. The main findings about the evolution of these relatively massive galaxies are: 1) There is evidence that, for both SDSS ans zCOSMOS galaxies, the mean specific SFR within a given population (either disk-dominated or bulge-dominated) is independent of SigmaM; 2) The observed SSFR - SigmaM step-function relation is due, at all investigated redshifts, to the changing mix of disk-dominated and bulge-dominated galaxies as surface density increases and the strong difference in the average SSFR between disks and bulges. We also find a modest differential evolution in the size-mass relations of disk and spheroid galaxies; 3) The shape of the median SSFR - SigmaM relation is similar, but with median SSFR values that are about 5-6 times higher in zCOSMOS galaxies than for SDSS, across the whole range of SigmaM, and in both spheroid and disk galaxies. This increase matches that of the global SFRD of the Universe as a whole, emphasizing that galaxies of all types are contributing, proportionally, to the global increase in SFRD in the Universe back to these redshifts (abridged).
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