A moving boundary problem describing the growth of a droplet in its vapour

In [1-5] and [12] the theory of shells is generalized: nonmaterial bidimensional continua re introduced in order to model capillarity phenomena. In this paper we solve some mathematical problems arising when the quoted models are used to describe the growth in its vapour of a sufficiently small drop in the neighbourhood of an equilibrium state. We start to consider the source terms appearing in the integro-differential parabolic evolution equation (IDE) deduced in [5] for the temperature field in the vapour phase. We prove that, due to coupling between the capillarity and thermomechanical phenomena occurring close to the interface, these terms have both space and time Holder coefficients equal to that onerelative to the second time derivative of the vradius of the droplet. T our knowledge, only GEVREY [14] partially treated this case for PDE of parabolic type. We improve his results in order to prove the well-posedness of the moving boundary problem formulated in [5] for IDE.