The time evolution of giant lobe-dominated radio galaxies (with projected linear size D>1 Mpc if H_{0}=50 km/s/Mpc and q_{0}=0.5 is analysed on the basis of dynamical evolution of the entire FRII-type population. Two basic physical parameters, namely the jet power Q_{0} and central density of the galaxy nucleus rho0 are derived for a sample of giants with synchrotron ages reliably determined, and compared with the relevant parameters in a comparison sample of normal-size sources consisting of 3C, B2, and other sources. Having the apparent radio luminosity P and linear size D of each sample source, Q_{0} and rho_{0} are obtained by fitting the dynamical model of Kaiser et al. (1997). We find that: (i) there is not a unique factor governing the source size; they are old sources with temperate jet power (Q_{0}) evolved in a relatively low-density environment (rho_{0}). The size is dependent, in order of decreasing partial correlation coefficients, on age; then on Q_{0}; next on rho_{0}. (ii) A self-similar expansion of the sources cocoon seems to be feasible if the power supplied by the jets is a few orders of magnitude above the minimum-energy value. In other cases the expansion can only initially be self-similar; a departure from self-similarity for large and old sources is justified by observational data of giant sources. (iii) An apparent increase of the lowest internal pressure value observed within the largest sources cocoon with redshift is obscured by the intrinsic dependence of their size on age and the age on redshift, which hinders us from making definite conclusions about a cosmological evolution of intergalactic medium (IGM) pressure.
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