Vacuum isolating, blow up threshold, and asymptotic behavior of solutions for a nonlocal parabolic equation

In this paper, we consider a nonlocal parabolic equation associated with initial and Dirichlet boundary conditions. First, we discuss the vacuum isolating behavior of solutions with the help of a family of potential wells. Then we obtain a threshold of global existence and blow up for solutions with critical initial energy. Furthermore, for those solutions that satisfy J(u(0)) <= d and I(u(0)),not equal 0, we show that global solutions decay to zero exponentially as time tends to infinity and the norm of blow-up solutions increases exponentially. Published by AIP Publishing.