  We consider the evolution of a $n$-dimensional convex hypersurface in the
euclidean space under mean curvature flow with densities $e^{\varepsilon
\frac12 n\mu^2 |x|^2}$, $\varepsilon =\pm 1$, and completely determine it
depending on the relation between $\mu$ and the upper or lower bound of the
normal curvatures of the evolving hypersurface at time 0.
