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: ut=Δu+|u|p−1u−1|Ω|∫Ω|u|p−1udx,x∈Ω,t>0, under homogeneous Neumann boundary condition in a bounded domain Ω⊂Rn,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+2n−2.