Powers of sums and their associated primes

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.