  We study the $E_1$-degeneration of the logarithmic Hodge to de Rham spectral
sequence of the special fiber of a semistable family over a discrete valuation
ring. On the one hand, we prove that the $E_1$-degeneration property is
invariant under admissible blow-ups. Assuming functorial resolution of
singularities over $\mathbb{Z}$, this implies that the $E_1$-degeneration
property depends only on the generic fiber. On the other hand, we show by
explicit examples that the decomposability of the logarithmic de Rham complex
is not invariant under admissible blow-ups, which answer negatively an open
problem of L. Illusie (Problem 7.14 \cite{Illusie2002}). We also give an
algebraic proof of an $E_1$-degeneration result in characteristic zero due to
Steenbrink and Kawamata-Namikawa.
