Blow-up for a semilinear parabolic equation with large diffusion on RN

We consider the Cauchy problem for a semilinear heat equation,{∂tu=DΔu+|u|p−1u,x∈RN,t>0,u(x,0)=λ+φ(x),x∈RN, where D>0, p>1, N⩾3, λ>0, and φ∈L∞(RN)∩L1(RN,(1+|x|)2dx). In this paper we assume∫RNφ(x)dx>0, and study the blow-up time and the location of the blow-up set of the solution for the case where D is sufficiently large. In particular, we prove that the location of the blow-up set depends on the large time behavior of the hot spots for the heat equation.