Blow-up phenomena for a nonlocal semilinear parabolic equation with positive initial energy

This paper is concerned with the blow-up of solutions to the following semilinear parabolic equation: u(t) = Delta u + vertical bar u vertical bar(p-1)u - 1/vertical bar Omega vertical bar integral(Omega) vertical bar u vertical bar(p-1)u dx, x is an element of Omega, t > 0, under homogeneous Neumann boundary condition in a bounded domain Omega subset of R-n, n >= 1, with smooth boundary. For all p > 1, we prove that the classical solutions to the above equation blow up in finite time when the initial energy is positive and initial data is suitably large. This result improves a recent result by Gao and Han (2011) which asserts the blow-up of classical solutions for n >= 3 provided that 1 < p <= n+2/n-2. (C) 2015 Elsevier Ltd. All rights reserved.