  For a proper, flat, generically smooth scheme $X$ over a complete DVR with
finite residue field of characteristic $p$, we define a specialization morphism
from the rigid cohomology of the geometric special fibre to $D_{crys}$ of the
$p$-adic \'etale cohomology of the geometric generic fibre, and we make a
conjecture ("$p$-adic local invariant cycle theorem") that describes the
behavior of this map for regular $X$, analogous to the situation in $l$-adic
\'etale cohomology for $l\neq p$. Our main result is that, if $X$ has
semistable reduction, this specialization map induces an isomorphism on the
slope $[0,1)$-part.
