  Multidimensional cosmological models with a higher dimensional space-time
manifold are investigated under dimensional reduction. In the Einstein
conformal frame, the effective potential for the internal scale factors is
obtained. The stable compactification of the internal spaces is achieved due to
the Casimir effect. In the case of more than one internal space a Casimir-like
ansatz for the energy density of the massless scalar field fluctuations is
proposed. Stable configurations with respect to the internal scale factor
excitations are found in the cases of one and two internal spaces.
