Invariant sets and the blow up threshold for a nonlocal equation of parabolic type

In this paper, we consider a nonlocal equation of parabolic type on a bounded smooth domain with Dirichlet boundary condition. First, the well-posedness in Lebesgue spaces is established by using the Banach fixed point theorem. Then we set up the regularity of the solution. More interestingly, we find the invariant sets for the initial boundary value problem and derive a threshold of blow up and global existence for its solution.