  The aim of this paper is to study certain properties of the weight spectral
sequences of Rapoport-Zink by a specialization argument. By reducing to the
case over finite fields previously treated by Deligne, we prove that the weight
filtration and the monodromy filtration defined on the $l$-adic \'etale
cohomology coincide, up to shift, for proper smooth varieties over equal
characteristic local fields. We also prove that the weight spectral sequences
degenerate at $E_2$ in any characteristic without using log geometry. Moreover,
as an application, we give a modulo $p>0$ reduction proof of a Hodge analogue
previously considered by Steenbrink.
