Blow-up on the boundary: a survey

Abstract

where m, p > 0 and Ω is either a smoothly bounded domain in R or Ω = R+ = {(x1, x′) : x′ ∈ RN−1, x1 > 0}, ν is the outward normal. Over the past two decades this problem has received considerable interest. For Ω bounded, m = 1 and p > 1 it was shown by Levine and Payne ([LP1]) in 1974 and by Walter ([Wa]) in 1975 that there are solutions which blow up in finite time. This means that lim sup t→T max Ω u(x, t) =∞ for some T <∞.