  In this paper, we will prove subadditivity of Kodaira dimensions for a
fibration with possibly singular geometric generic fiber, under certain nefness
and relative semi-ampleness conditions. As an application, for a fibration $f:
X \to Y$ of a smooth projective threefold over an algebraically closed field of
characteristic $p>5$, under the assumption that $Y$ is of general type and
non-uniruled, we prove subadditivity of Kodaira dimensions when general fibers
are smooth or when $K_{X/Y}$ is relatively big over $Y$.
