Neutron reflectivity has been used to measure concentration profiles of polymer brushes as they penetrate a high molecular weight polymer matrix. The brushes consist of end-functionalized deuterated polystyrene strongly grafted to a silicon substrate, in a melt matrix of (protonated) polystyrene. Initially the brush is totally excluded from the matrix. The rate at which the brush chains penetrate the matrix was investigated by measuring the brush segment density profiles after annealing the samples above the glass transition temperature for a range of times. We find that when the initial interface between the brush and the matrix is sharp, there is at first a rapid penetration of the brush into the matrix. This is followed by a slow approach to equilibrium, which is not achieved after annealing for a day. These observations are consistent with the predictions of a recent analytical theory and computer simulations. In contrast, samples with an initially diffuse interface reach equilibrium quickly. Two different matrix polymers were used to investigate the effect of the matrix molecular weight. Comparison has been made with a mean field model for the equilibrium brush profile.