Some recent results on blow-up on the boundary for the heat equation

Abstract

(1.3) u(x, 0) = u0(x), x ∈ Ω, where p > 1 and Ω is either a smoothly bounded domain in R or Ω = R+ := {x ∈ R : xN > 0}, ν is the outward normal. Some closely related problems are also considered. This paper is a sequel of the previous survey [FF] therefore we focus our attention on results that have been established after [FF] appeared. For Ω bounded it was shown by Levine and Payne ([LP1], [LP2]) in 1974 and by Walter ([W]) in 1975 that there are solutions which blow up in finite time. This means