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We investigate modification of Kolmogorov wave turbulence in QCD calculatinggluon spectra as functions of time in the presence of a low energy source whichfeeds in energy density in the infrared region at a time-dependent rate. Thenconsidering the picture of saturation constraints as has been constructed inthe "bottom-up" thermalization approach we revisit that picture for RHICcenter-mass energy, $W = 130 {GeV}$, and also extend it to LHC center-massenergy, $W = 5500 {GeV}$, thus for two cases having an opportunity to calculatethe equilibration time, $tau_{eq|therm}$, of the gluon system produced in acentral heavy ion collision at mid-rapidity region. Thereby, at RHIC and LHCenergies we can match the equilibration time, obtained from the late stagegluon spectrum of the modified Kolmogorov wave turbulence, onto that of the"bottom-up" thermalization and other evolutional approaches as well. Inaddition, from the revised "bottom-up" approach we find the gluon liberationcoefficient to be on the average, $c = 0.81 - 1.06$ at RHIC and $c = 0.50 -0.56$ at LHC. We also present other phenomenological estimates of $tau_{therm}$which, at QCD realistic couplings, yield $0.45fm - 0.65fm < tau_{therm} <0.97fm - 2.72fm$ at RHIC and $0.31fm - 0.40fm < tau_{therm} < 0.86fm - 2.04fm$at LHC, both reflecting the original and modified Kolmogorov wave turbulentscenarios. In the latter case, at certain conditions, taking also into accountboth very small and realistic couplings we give estimates - $0.65fm <tau_{therm} < 1.29fm$ at RHIC and $0.52fm < tau_{therm} < 1.16fm$ at LHC, aswell as at realistic couplings we find $0.53 < tau_{therm} < 0.7fm$ at RHIC and$0.41 < tau_{therm} < 0.65fm$ at LHC.

Modified Kolmogorov Wave Turbulence in QCD matched onto "Bottom-up" Thermalization