Blow-up problems for a semilinear heat equation with large diffusion

We consider the blow-up problem of a semilinear heat equation, u_t = DΔu + u^p in Ω × (0, T_D), ∂u/∂ν(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. We study the blow-up time, the location of the blow-up set, and the blow-up profile of the blow-up solution for sufficiently large D. In particular, we prove that, for almost all initial data φ, if D is sufficiently large, then the solution blows-up only near the maximum points of the orthogonal projection of the initial data φ from L² (Ω) onto the second Neumann eigenspace.