Blow-up problems for a semilinear heat equation with large diffusion

We consider the blow-up problem of a semilinear heat equation,ut=DΔu+upinΩ×(0,TD),∂u∂ν(x,t)=0on∂Ω×(0,TD),u(x,0)=φ(x)⩾0inΩ,where Ω is a bounded smooth domain in RN, TD>0, D>0, and p>1. We study the blow-up time, the location of the blow-up set, and the blow-up profile of the blow-up solution for sufficiently large D. In particular, we prove that, for almost all initial data φ, if D is sufficiently large, then the solution blows-up only near the maximum points of the orthogonal projection of the initial data φ from L2(Ω) onto the second Neumann eigenspace.