抄録
This paper is concerned with positive solutions of semilinear diffusion equations ut=ε2\ riangle 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(overline{Ω}) has a blow-up set S0=\displaystyle {x∈overline{Ω}|u0(x)=max_{y∈overline{Ω}}u0(y)} and a blow-up profile \[u*0(x)=(u0(x)-(p-1)-(max_{y∈overline{Ω}}u0(y))-(p-1))-1/(p-1) \] outside the blow-up set S0. For the diffusion equation ut=ε2\ riangle u+up in Ω under the boundary condition ∂ u/∂ v=0 on ∂Ω, it is shown that if a positive function u0∈ C2(overline{Ω}) satisfies ∂ u0/∂ v=0 on ∂Ω, then the blow-up profile u*ε(x) of the solution uε with initial data u0 approaches u*0(x) uniformly on compact sets of overline{Ω}\backslash S0 as ε→+0.
