The thermal structure of the vertical boundary layers in mantle convection can be seriously influenced by the phonon contribution of the thermal conductivity because of its decreasing nonlinear dependence with increasing temperature. Such a dependence would induce the cold descending slabs to be considerably warmer than the canonical models with constant conductivity. We have carried out both 2-D and 3-D convection calculations, using a temperature and pressure-dependent thermal conductivity with a jump at 670 km depth. In order to determine the influence of advection in counteracting the nonlinear thermal diffusion, we have gone up to a Rayleigh number of 7¿106 for a constant viscosity model with variable conductivity. Our comparison shows unequivocally that the cold downwellings with a high conductivity are disappearing to a greater degree in the lower mantle at depths of around 1500 to 2000 km than models with constant conductivity. This divergence in the visibility of cold downwellings increases with larger Rayleigh numbers, because of the negative feedback nature of the nonlinear dependence in the temperature-dependent conductivity. Our results would suggest that many cold slabs sinking through the lower mantle with a realistic conductivity would become thermally assimilated in the bottom 1000 km of the mantle. ¿ 2000 American Geophysical Union