On some nonlocal parabolic systems with gradient source terms

The present paper is concerned with a class of nonlocal parabolic systems set in a regular bounded open subset of R N , where the gradients of the unknowns act as source terms. First, we establish some nonexistence and blow-up in finite time results. Second, we prove some new weighted regularity results. Such results are interesting in themselves and play a crucial role to study local existence of nonnegative solutions under suitable assumptions on the data. This work also highlights a substantial difference between the nonlocal case and the local case already studied by the fourth author and his coworkers. Mathematics Subject Classification 2020: 35R11 . 35J62 . 30G50 . 47H10 . 35B45.