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

Abstract

This paper is concerned with positive solutions of semilinear diffusion equations ut=ε2∆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)=maxy∊Ω¯u0(y)} and a blowup profile u*0(x)=(u0(x)-(p-1)-(maxy∊Ω¯u0(y))-(p-1))-1/(p-1) outside the blow-up set S0. For the diffusion equation ut=ε2∆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 u*ε(x) of the solution uε with initial data u0 approaches u*0(x) uniformly on compact sets of Ω¯∖S0 as ε→+0.