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