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In this paper, we study an algebraic fiber space in positive characteristicwhose generic fiber $F$ has finitely generated canonical ring and sufficientlylarge 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 aFujita-type conjecture due to Popa and Schnell concerning direct images ofpluricanonical bundles, and prove it under some additional hypotheses. As anapplication, we show the subadditivity of Kodaira dimensions in some new cases.

Direct images of pluricanonical bundles and Frobenius stable canonical rings of fibers