Approaching a vertex in a shrinking domain under a nonlinear flow

Abstract

We consider here the homogeneous Dirichlet problem for the equation u(t)= uΔu - γ|∇u|(2) with γ ∈ R, u ≥ 0, in a noncylindrical domain in space-time given by |x| ≤ R(t) = (T - t)(p), with p > 0. By means of matched asymptotic expansion techniques we describe the asymptotics of the maximal solution approaching the vertex x = 0, t = T, in the three different cases p > 1/2, p = 1/2(vertex regular), p < 1/2 (vertex irregular).