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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.

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In this paper we show how to obtain wax diffusivity and solubility in crude oils from deposition measurements in the cold finger device with stirring. We present a mathematical model in which the physical quantities are assumed to be space-independent in the bulk region of the device, because of agitation. Comparison with available laboratory results is provided, showing that the wax diffusivity values obtained in simulations are in agreement with data of field experiments.