A Stefan Problem for a Reaction-Diffusion System
Avner Friedman, DAVID S. ROSS, Jianhua Zhang · SIAM Journal on Mathematical Analysis · 1995
The paper deals with a Stefan problem for a system of three weakly coupled semi-linear parabolic equations. The system describes dissolution of a spherical particle in solution. The dissolved species A reacts chemically with species B already in the solution, thereby forming species C. Species C diffuses in the solution and some of it adsorbs to the particle’s boundary and gradually shuts down the dissolution. It is shown that the mathematical model has a unique solution with finite shut-down time. When the reaction rate K increases to infinity, the limit model should exhibit phase separation between A and B, and it thus has two free boundaries; the particle’s boundary and the $A - B$ interface. It is proved, in the case in which A and B diffuse at the same rate, that the solution with finite K converges to the solution of the limit problem, and the A phase in the limit problem disappears in finite time.