Moduli instability in warped compactifications of the type-IIB supergravity

We show that the conifold and deformed-conifold warped compactifications of the ten-dimensional type-IIB supergravity, including the Klebanov-Strassler solution, are dynamically unstable in the moduli sector representing the scale of a Calabi-Yau space, although it can be practically stable for a quite long time in a region with a large warp factor. This instability is associated with complete supersymmetry breaking except for a special case and produces significant time-dependence in the structure of the four-dimensional base spacetime as well as of the internal space.