### Parabolic Free Boundary Problems in Industrial and Biological Applications

#### Abstract

We present a collection of mathematical models of processes in which diffusion is an important transport mechanism and therefore governed by partial differential equations of parabolic type (or of elliptic type in the steady state). Diffusion is ubiquitous in nature and many other processes can be modelled by differential equations having the same mathematical character. The feature representing the main common denominator of all the problems here considered is the fact that the domain where the differential governing system is to be solved is partially unknown. Any part of the boundary to be determined is called a free boundary. Problems of this kind are encountered very frequently in applications. The variety of physical and mathematical situations is impressively large. One of our aims is indeed to give an idea of the broadness of this subject, but the main scope is to train the reader to formulate a mathematical model, trying to catch the main elements and at the same time to state the applicability limits generated by the approximations that are introduced in the course of the formulation. Such a plan is obviously rather ambitious, since for each case it requires an illustration on the physical background and the analysis of the peculiar mathematical aspects. In order to make the book usable for didactical purposes conciseness is necessary. For this reason our exposition will be synthetic and generally confined to the more representative cases, referring to the specific literature for more details. A background in partial differential equations is clearly of great help, but the main results on parabolic are summarized to make these notes accessible to non-mathematicians.

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