Based on corrected rate equations, the kinetics is revealed for the initial growth of thin films with a low nucleus density and a linear lateral growth rate on a substrate surface. In this case, the mobility and coalescence terms in the rate equations are neglected, and the coverage term takes a dominant role. All the calculated results are in agreement with experimental data. These results introduce a new scaling in the case of dominant coverage,which is different from the those in the case of coalescence or mobility in a dominant role.