Mathematical results for a model of diffusion and precipitation of chemical elements in solid matrices

This paper discusses mathematical results for a classical model of simultaneous diffusion and precipitation of chemical elements in solid matrices. This model consists of diffusion equations coupled with the laws of mass action, which express the hypothesis of instantaneous local thermodynamic equilibrium between the matrix and precipitate phases. Existence and uniqueness of the solution of the local problem of thermodynamic equilibrium are established, but the full boundary value problem is shown to be prone to severe mathematical pathologies which are interpreted as limitations of the model due to some oversimplifying physical assumptions. An improved version of the model is proposed and shown to fit within the framework of Amann's theory for systems of strongly coupled, quasilinear equations of parabolic type; Amann's results ensure existence, uniqueness and smoothness of maximal solutions. It is also shown that positivity of solutions is ensured provided that it is guaranteed initially and on the boundary of the domain. However, the problem of global existence of a solution remains open.