Approaching a vertex in a shrinking domain under a nonlinear flow

Abstract

We consider here the homogeneous Dirichlet problem for the equation \( u_t = u\Delta u - \gamma |\nabla u|^2 \quad \mathrm{with} \quad \gamma \in R, u \geq 0 \) , in a noncylindrical domain in space-time given by \( |x| \leq R(t) = (T - t)^p, \quad \mathrm{with} \quad p > 0 \). 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).