Singularity formation in the one-dimensional supercooled Stefan problem

Abstract

It is well-known that solutions to the one-dimensional supercooled Stefan problem (SSP) may exhibit blow-up in finite time. If we consider (SSP) in a half-line with zero flux conditions at t = 0, blow-up occurs if there exists T < ∞ such that limt↑T s(t) > 0 and lim inft↑T(t) = – ∞,s(t) being the interface of the problem under consideration. In this paper, we derive the asymptotics of solutions and interfaces near blow-up. We shall also use these results to discuss the possible continuation of solutions beyond blow-up.