On some fractional parabolic reaction-diffusion systems with gradient source terms

The present paper is concerned with a fractional parabolic reaction-diffusion system posed in a regular bounded open subset of \({\mathbb {R}}^N\), where the gradients of the unknowns act as source terms (see (S) below). 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 to our system 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.