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

Abstract This paper is concerned with the blow-up of solutions to the following semilinear parabolic equation: u t = Δ u + | u | p − 1 u − 1 | Ω | ∫ Ω | u | p − 1 u d x , x ∈ Ω , t > 0 , under homogeneous Neumann boundary condition in a bounded domain Ω ⊂ 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 .