  (Abridged) To define a framework for the formation and evolution of the
cooling cores in X-ray galaxy clusters, we study how the physical properties
change as function of the cosmic time in the inner regions of a 4 keV and 8 keV
galaxy cluster under the action of radiative cooling and gravity only. The
cooling radius, R_cool, defined as the radius at which the cooling time equals
the Universe age at given redshift, evolves from ~0.01 R200 at z>2, where the
structures begin their evolution, to ~0.05 R200 at z=0. The values measured at
0.01 R200 show an increase of about 15-20 per cent per Gyr in the gas density
and surface brightness and a decrease with a mean rate of 10 per cent per Gyr
in the gas temperature. The emission-weighted temperature diminishes by about
25 per cent and the bolometric X-ray luminosity rises by a factor ~2 after 10
Gyrs when all the cluster emission is considered in the computation. On the
contrary, when the core region within 0.15 R500 is excluded, the gas
temperature value does not change and the X-ray luminosity varies by 10-20 per
cent only. The cooling time and gas entropy radial profiles are well
represented by power-law functions. The behaviour of the inner slopes of the
gas temperature and density profiles are the most sensitive and unambiguous
tracers of an evolving cooling core. Their values after 10 Gyrs of radiative
losses, T_gas ~ r^0.4 and n_gas ~ r^(-1.2) for the hot (cool) object, are
remarkably in agreement with the observational constraints available for nearby
X-ray luminous cooling core clusters. Because our simulations do not consider
any AGN heating, they imply that the feedback process does not greatly alter
the gas density and temperature profiles as generated by radiative cooling
alone.
