We present the evolution of the comoving SFR density in the redshift range 0 < z < 5 using the first epoch data release of the VVDS, that is 11564 spectra selected at I_AB=24 over 2200 arcmin^2 in two fields of view, the VVDS-0226-04 and the VVDS-CDFS-0332-27, and the cosmological parameters (Omega_M, Omega_L, h)=(0.3, 0.7, 0.7). We study the multi-wavelength non dust-corrected luminosity densities at 0 < z < 2 from the rest-frame FUV to the optical passbands, and the rest-frame 1500A luminosity functions and densities at 2.7 < z < 5. They evolve from z=1.2 to 0.05 according to (1+z)^{x} with x = 2.05, 1.94, 1.92, 1.14, 0.73, 0.42, 0.30 in the FUV-1500, NUV-2800, U-3600, B-4400, V-5500, R-6500, and I-7900 passbands, respectively. From z = 1.2 to 0.2 the B-band density for the irregular-like galaxies decreases markedly by a factor 3.5 while it increases by a factor 1.7 for the elliptical-like galaxies. We identify several SFR periods; from z = 5 to 3.4 the FUV-band density increases by at most 0.5dex, from z=3.4 to 1.2 it decreases by 0.08dex, from z=1.2 to 0.05 it declines steadily by 0.6dex. For the most luminous M_AB(1500) < -21 galaxies the FUV-band density drops by 2dex from z = 3.9 to 1.2, and for the intermediate -21 < M_AB(1500) < -20 galaxies it drops by 2dex from z = 0.2 to 0. Comparing with dust corrected surveys, at 0.4 < z < 2 the FUV seems obscured by a constant factor of ~1.8-2 mag, while at z < 0.5 it seems progressively less obscured by up to ~0.9-1 mag when the dust-deficient early-type population is increasingly dominating the B-band density. The VVDS results agree with a downsizing picture where the most luminous sources cease to efficiently produce new stars 12 Gyrs ago (at z~4), while intermediate luminosity sources keep producing stars until 2.5 Gyrs ago (at z~0.2).(abridged)
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