Global existence and blow-up of solutions to a nonlocal parabolic equation with singular potential

Abstract

In this paper, we study a nonlocal parabolic equation with singular potential on a bounded smooth domain with homogeneous Neumann boundary condition. Firstly, we find a threshold of global existence and blow-up to the solutions of the problem when the initial data is at the low energy level, i.e., J(u0)≤d, where J(u0) is the initial energy and d is the mountain-pass level. Moreover, when J(u0)<d, the vacuum isolating behavior of the solutions is also discussed. Secondly, we prove that there exist solutions of the problem with arbitrary initial energy that blow up in finite time. We also obtain the upper bounds of the blow-up time for blow-up solutions.