One of major results due to Ogus-Vologodsky in the nonabelian Hodge theory over positive characteristic is the generalization of Decomposition Theorem of Deligne-Illusie to the case with coefficients. In this talk, I am going to report our recent work on the decomposition theorem with intersection condition, which generalizes the previous decomposition theorems in a natural way. As application, we prove that the Hodge spectral sequence of the intersection de Rham complex, attached to a strict $p$-torsion logarithmic Fontaine module, degenerates at $E_1$. This is a joint work with Zhang Zebao.