A critical case for the solvability of stefan‐like problems

Abstract

We consider a one‐phase one‐dimensional Stefan problem with general data with the aim to investigate some open questions on existence of classical solutions. We show how existence and nonexistence are discriminated by the behavior of the initial datum in the neighborhood of the starting point of the free boundary.

Some remarks on the regularization of supercooled one-phase Stefan problems in one dimension

“…Then the fact that lim^_>Xo+ u{x, to) < 0 implies that a{t) cannot be bounded in the vicinity of t = to-□ (f) The solution of (4.1) exists. Since by assumption c(x, to) > 0 for Xo < x < s(t0), there must be points in a small neighborhood (xo,Xo + £) *n w^ich u(x,to) > -1, where u{x,t0) coincides with lim/o_ u(x,t),u solving (SSP) for t < to-Using once more Proposition 3.2 and [24], existence and uniqueness are guaranteed. □ If Xo = 0 we modify the boundary conditions at x = 0 for / > t0 introducing a waiting time r as described in Sec.…”

“…By virtue of Proposition 3.2 we can assert that u(x,t*) < -1 in the whole interval for e sufficiently small. At this point, nonexistence for t > t* follows from [24].…”

“…The proof of the converse statement is also easy: if in (s(t*),t*) we have essential blow-up, then the inequality u(x,t*) > -1 cannot be satisfied in any interval (.s(t*) -e, s(t*)), otherwise the solution would be continuable [24], Then, by Proposition 3.2 c < 0, c ^ 0 for any e sufficiently small. □ It is of some importance to distinguish the role played in the theory by the set of inner points where c = 0 and the set of the points where u = -1.…”

On the stability of some solutions of the Stefan problem

“…The problem is then the following In this paper we will assume that (j>{x) > -1. This condition ensures that the problem has a solution for any time / > 0, (see Fasano & Primicerio 1983). This problem has two types of special solutions.…”

The Supercooled One-Phase Stefan Problem in Spherical Symmetry

“…Problems of this kind have been studied by other authors in connection with the freezing of a supercooled liquid. Several different boundary conditions were analysed in [3], [5], [6], [7], [10], [11], [13], for the one-dimensional case, in [1], [2] for cylindrical symmetry and in [9] for spherical symmetry.…”