Location of blow-up set for a semilinear parabolic equation with large diffusion

This paper is concerned with {u_t = DΔu + u^p in Ω × (0, T_D), (P) ∂u/∂v (x, t) = 0 on ∂Ω × (0, T_D), u (x, 0) = φ(x) ≥ 0 in Ω, where Ω is a bounded smooth domain in R^N, T_D > 0, D > 0, and p > 1 with (N-2) p ≤ N+2. Let P₂ be the projection from L²(Ω) onto the second Neumann eigenspace. We prove that, if P₂φ ≢ 0 in Ω and D is sufficiently large, the solution u of (P) blows up only near the set M ∪ ∂Ω, where script M sign = {x ∈ Ω̄: (P₂φ)(x) = max_y∈Ω̄(P₂φ)(y)}.