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Abstract

Abstract. We study different initial and boundary value problems for the onedimensional heat-diffusion equation, with the purpose of establishing conditions on initial and boundary data that ensure the solution is, during a certain interval of time, bounded by two predetermined constants. When the constants represent phasechange temperatures of a material medium, the desired conditions can be physically interpreted by saying that the medium does not undergo a phase-change during a certain lapse of time; i.e., the heat-conduction model will preserve its validity in the meantime. The tools employed to tackle this kind of problem consist in reducing the initial and boundary value problems to a Volterra integral equation. For these equations there exist simple methods to estimate their solutions. We improve some results that appeared in [8].1. Introduction and preliminaries. An essential part of the construction of mathematical models of a physical phenomenon is to specify their scope. Particularly, once a model of some phenomenon of evolution has been formulated, it is natural to attempt to determine its temporary range of validity. In the case of models of heat-conduction in material media, an important limitation of this range is imposed by the change of phase phenomena. Hence the necessity of modifying the model to include this characteristic occurs: free-boundaries and mushy regions could now appear. It is then desirable to establish conditions on initial and boundary data of the problem so that phase-changes of the medium during a given time cannot occur.These theoretical considerations have an immediate practical counterpart: when we operate a thermal engine (think of a blast furnace or a nuclear reactor), it is necessary to fix conditions of operation that ensure that the temperature in certain parts of the engine never exceeds the fusion point of the constitutive material. More often, we wish to operate in a range of temperatures for which the process is, in some sense, optimal. On the other hand, in diffusion processes, we may wish that an