  Let $A, B$ be polynomial rings over a field $k$, and $I\subseteq A,
J\subseteq B$ proper homogeneous ideals. We analyze the associated primes of
powers of $I+J\subseteq A\otimes_k B$ given the data on the summands. The
associated primes of large enough powers of $I+J$ are determined. We then
answer positively a question about the persistence property of $I+J$ in many
new cases.
