The birth of a cusp in the two-dimensional, undercooled Stefan problem

This paper deals with the one-phase, undercooled Stefan problem, in space dimension $N = 2$. We show herein that planar, one-dimensional blow-up behaviours corresponding to the undercooling parameter $\Delta = 1$ are unstable with respect to small, transversal perturbations. The solutions thus produced are shown to generically generate cusps in finite time, when they exhibit an undercooling $\Delta = 1 - O\left ( \epsilon \right ) < 1$, where $0 < \epsilon << 1$, and $\epsilon$ is a parameter that measures the strength of the perturbation. The asymptotic behaviour of solutions and interfaces near their cusps is also obtained. All results are derived by means of matched asymptotic expansions techniques.