On some nonlocal parabolic systems with gradient source terms

Abstract

The present paper is concerned with a class of nonlocal parabolic systems set in a regular bounded open subset of RN, 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.

On Some Nonlocal Elliptic Systems with Gradient Source Terms

“…In a forthcoming work [8], we will consider a class of generalized systems with different diffusions, for instance:…”

“…where s 1 ̸ = s 2 . Nonlocal parabolic systems with local gradient terms will also be considered in [9].…”