Blow-up profile of a solution for a nonlinear heat equation with small diffusion

This paper is concerned with positive solutions of semilinear diffusion equations u_t = ε² Δ u + u^p in Ω with small diffusion under the Neumann boundary condition, where p > 1 is a constant and Ω is a bounded domain in R^N with C² boundary. For the ordinary differential equation u_t = u^p, the solution u⁰ with positive initial data u⁰ ∈ C(Ω) has a blow-up set S⁰ = (x ∈ Ω|u₀(x) = max_yeΩ u₀(y)) and a blowup profile u⁰_*(x)=(u₀(x)^-(p-1) - (max_yeΩ u₀(y))^-(p-1))^-1/(p-1) outside the blow-up set S⁰. For the diffusion equation u_t = ε² Δ u + u^p in Ω under the boundary condition ∂u/∂v = 0 on ∂Ω, it is shown that if a positive function u₀ ∈ C²(Ω) satisfies ∂u₀/∂v = 0 on ∂Ω, then the blow-up profile u^ε_*(x) of the solution u^ε with initial data u₀ approaches u⁰_*(x) uniformly on compact sets of Ω \ S⁰ as ε → +0.