  In this paper, we study an algebraic fiber space in positive characteristic
whose generic fiber $F$ has finitely generated canonical ring and sufficiently
large Frobenius stable canonical ring. An example of such a case is when $F$ is
$F$-pure and its dualizing sheaf is invertible and ample. We treat a
Fujita-type conjecture due to Popa and Schnell concerning direct images of
pluricanonical bundles, and prove it under some additional hypotheses. As an
application, we show the subadditivity of Kodaira dimensions in some new cases.
