New results on some classical parabolic free-boundary problems

Abstract

With an appendix by A. LACEY (New College, Oxford) 1. Introduction. In [1] we analyzed a class of free-boundary problems for the heat equation in one space dimension, releasing the sign restrictions on the data and the latent heat usually required in the Stefan problem. Problems of this kind have been studied by other authors also in connection with the freezing of a supercooled liquid and with decision theory (see e.g. [2]- [16], and the references quoted herein).Two major problems remain open or not completely solved, namely (i) does any solution exist when the datum prescribed on the free boundary x = s(t) does not fit the initial datum at x = s(0)?; (ii) how are the data related to the possibility of continuing the solution in arbitrarily large time intervals?Sec. 2 of this paper contributes toward answering the above questions. Special results which are scattered in the literature cited can be found in the framework of our analysis (sometimes with relevant simplifications of the arguments).It is known that some free-boundary problems with the Cauchy data prescribed on the free boundary can be reduced to schemes of the type mentioned above, provided that suitable compatibility conditions are fulfilled by the data. A typical example in which such conditions are violated is given by the diffusion-consumption of oxygen in insulated living tissues, when the initial oxygen distribution coincides with the steady-state profile corresponding to a given constant input (see [14] and [17]-[23]).This case is considered in Sec. 3, there we prove the existence of a smooth solution and remark that the associated problem for the time derivative of the oxygen concentration is of the type considered in Sec. 2 but with an initial datum behaving like a "5-function " at the origin. A very sharp estimate of the lifetime of the tissue is also obtained by means of elementary calculations.In Sec. 4 we prove some comparison theorems for the solutions of the problems dealt with in the preceding sections.As an interesting consequence, a nonexistence theorem will identify a class of initial data such that the answer to question (i) above is negative.

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

“…The first statement is proved in [5], The second result follows from the fact that as long as 5 is bounded the solution of (SSP) is smooth in the vicinity of the free boundary. □…”

“…On the basis of the results of [5] we can say that the symmetric left and right free boundaries for (CDCP) with initial data (p have to meet in a finite time. Hence the same has to happen to the curves x = a(t),x = s(t).…”

“…The aim is twofold. First we seek more information on the possible behaviour of the solution, complementing the analysis performed in [5]; then we shall use the results obtained to suggest a regularization of the model which avoids blow-up and describes phase change to completion.…”

“…(1.6) Jo In the special case in which the equation Uq(x) = -1 has no more than one root in [0,1] Q < 0 is necessary and sufficient for blow-up in the sense that T = t* < oo, s{t*) > 0 and lim,^r_ s(t) --oo; on the other hand Q > 0 and Q = 0 characterize global existence (T = +oo) and finite time extinction (,s(t) = 0), respectively [5]. In any event it is known [6] that if Q < 0 blow-up certainly occurs.…”

Sedimentation in Coal-Water Slurry Pipelining

“…When a C.W.S. is stirred in a vessel or pumped through a pipeline, the action of shear dramatically modifies this picture through a phenomenon called rheological degradation which has also been deeply investigated and fully explained (see, for example, [9], [10], [11], [15], [21], [22], [23], [24], [26], [32]). However the time scale of this phenomenon is many orders of magnitude larger than that of sedimentation, so that if we focus on the latter problem, the time dependency of rheological parameters can be neglected.…”