  Given arbitrary monomial ideals $I$ and $J$ in polynomial rings $A$ and $B$
over a field $K$, we investigate the Stanley depth of powers of the sum $I+J$,
and their quotient rings, in $A\otimes_K B$ in terms of those of $I$ and $J$.
Our results can be used to study the asymptotic behavior of the Stanley depth
of powers of a monomial ideal. For instance, we solved the case of monomial
complete intersection.
