This paper is concerned with positive solutions of semilinear diffusion equations ut = ε2 Δ u + up in Ω with small diffusion under the Neumann boundary condition, where p > 1 is a constant and Ω is a bounded domain in RN with C2 boundary. For the ordinary differential equation ut = up, the solution u0 with positive initial data u0 ∈ C(Ω) has a blow-up set S0 = (x ∈ Ω|u0(x) = maxyeΩ u0(y)) and a blowup profile u0*(x)=(u0(x)-(p-1) - (maxyeΩ u0(y))-(p-1))-1/(p-1) outside the blow-up set S0. For the diffusion equation ut = ε2 Δ u + up 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 uε*(x) of the solution uε with initial data u0 approaches u0*(x) uniformly on compact sets of Ω \ S0 as ε → +0.
!!!All Science Journal Classification (ASJC) codes
- 数学 (全般)