Blow-up set for a semilinear heat equation with small diffusion

We consider the blow-up problem for a semilinear heat equation,{∂tu=ϵΔu+upin Ω×(0,T),u(x,t)=0on ∂Ω×(0,T) if ∂Ω≠∅,u(x,0)=φϵ(x)⩾0in Ω, where Ω is a domain in RN, N⩾1, ϵ>0, p>1, and T>0. In this paper, under suitable assumptions on {φϵ}, we prove that, if the family of the solutions {uϵ} satisfies a uniform type I blow-up estimate with respect to ϵ, then the solution uϵ blows up only near the maximum points of the initial datum φϵ for any sufficiently small ϵ>0. This is proved without any conditions on the exponent p and the domain Ω, such as (N−2)p