The time evolution of giant lobe-dominated radio galaxies ( with projected linear size D > 1 Mpc if H _ { 0 } =50 km s ^ { -1 } Mpc ^ { -1 } 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 \rho _ { 0 } 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 .

Key words : galaxies : active – galaxies : evolution – galaxies : kinematics and dynamics