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

We are concerned with the Cauchy problem for a semilinear heat equation,(P){∂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 the paper of Fujishima and Ishige (2011) [8] the authors of this paper studied the behavior of the blow-up time and the blow-up set of the solution of (P) as D→∞ for the case ∫RNφ(x)dx>0. In this paper, as a continuation of Fujishima and Ishige (2011) [8], we consider the case∫RNφ(x)dx⩽0, and study the behavior of the blow-up time and the blow-up set of the solution of (P) as D→∞. The behavior in the case ∫RNφ(x)dx⩽0 is completely different from the one in the case ∫RNφ(x)dx>0.