Blow-up set for a semilinear heat equation and pointedness of the initial data

We consider the blow-up problem for a semilinear heat equation, (equation presented) where ε> 0, p> 1, N≥1, Ω is a domain in ℝ_N, and φ is a nonnegative smooth bounded function in Ω. It is known that, under suitable assumptions, if ε is sufficiently small, then the solution of (E) blows up only near the maximum points of the initial function φ (see, for example, [7]). In this paper, as a continuation of [7], we study the relationship between the location of the blow-up set and the level sets of the initial function φ. We also prove that the location of the blow-up set depends on the mean curvature of the graph of the initial function on its maximum points.