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 u_t=ε²\ riangle 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(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 u_t=ε²\ riangle u+u^p 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 u^ε with initial data u₀ approaches u_*⁰(x) uniformly on compact sets of overline{Ω}\backslash S⁰ as ε→+0.