DEMONSTRATIO MATHEMATICA Vol. XXXII No 1 1999 Paola Mannucci, Domingo Alberto Tarzia THE SUPERCOOLED ONE-PHASE STEFAN PROBLEM IN SPHERICAL SYMMETRY Abstract. The supercooled one-phase Stefan problem in spherical symmetry with a heat flux condition at the fixed face is considered. The relation between the heat flux and the initial temperature is analysed in order to characterize the cases with a global solution (possibility of continuing the solution for arbitrarily large time intervals), a finite time extinction and a blow-up at a finite time. 1. Introduction We study a supercooled one-phase Stefan problem in spherical symmetry (r G [T*O, 1], 7"o > 0) corresponding to a positive heat flux condition at the fixed face and a negative initial temperature. 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. In Section 1 we give the preliminaries corresponding to the description of the problem and in Section 2 we obtain conditions for data in order to characterize the cases with a global solution (possibility of continuing the solution for arbitrarily large time intervals), a finite time extinction and a blow-up at a finite time. In Section 3 we study the asymptotic behaviour of the solution and we give some results concerning the particular case 7*0 = 0 with null heat flux, which are a sequel to those given in [9]. In this paper we study the following problem: The first author is partially supported by the Italian M.U.R.S.T. National Project "Problemi non lineari..." Key words:One-phase Stefan problem, phase-change problem, parabolic free boundary problem. A MS Subject classifications: 35R35, 80A22, 35B40, 35K05.