A Stefan Problem for a Reaction-diffusion System

The paper deals with a Stefan problem for a system of three weakly coupled semilinear 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 di uses 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 nite shut-down time. When the reaction rate K increases to in nity, 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 di use at the same rate, that the solution with nite K converges to the solution of the limit problem, and that the A phase in the limit problem disappears in nite time. x1. The model. Consider a solid spherical particle composed of chemicalA with uniform concentration A . The particle is in a solution of chemical B. As the particle dissolves, the A that enters the solution reacts with B to form chemical C. Then C di uses in the solution and some of it reaches the solid particle and adsorbs to its surface. The presence of the adsorbed C inhibits the dissolution, and ultimately shuts it down entirely. Assuming radially symmetric data and radially symmetric functions A;B;C, we denote by r = R(t) the radius of the solid sphere at time t. Then the equations @A @t = DA A KAB ; (1.1) @B @t = DB B KAB ; (1.2) @C @t = DC C +KAB ; (1.3) hold in fr > R(t)g, where K is the reaction rate and DA;DB;DC are the di usion coe cients. These equations indicate that A and B are lost in a second-order reaction in which C is formed, and all three species di use. In the standard mass-action model of chemical kinetics, the concentrations are all expressed in moles/liter, and the coe cient K, the second-order reaction rate, is expressed in liters/(mole-sec). Then KAB is the number of moles per liter per second that undergo reaction; in our case, A and B are consumed, C is created, the same number of moles of A and B are lost, and this number of moles of C is created. A nice reference for this material is the book by Erdi and Toth [4]. Next, dR dt = @A @r on r = R(t) (1.4) where is a positive constant, i.e., the rate at which the radius of the particle decreases is proportional to the ux of species A away from the particle. We also have @B @r = 0 on r = R(t) ; (1.5) Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455, y Applied Mathematics and Statistics Group, Computational Science Laboratory, Eastman Kodak Company, Rochester, New York 14650{2205 z University of Minnesota, School of Mathematics, Minneapolis, Minnesota 55455 i.e., there is no ux of B through the particle's surface and B does not undergo any surface reaction. The adsorption of C to the surface is proportional to the local saturation; it is given by an empirical law DC@C=@r = C n for some positive constants , n (see [13, pp. 104{105]); for de niteness we take n = 4, that is, DC @C @r = C : (1.6) However, all the results of this paper remain valid with minor changes if we replace C by any other monotone increasing function f(C) with f(0) = 0; f(C) > 0 for C > 0. The boundary conditions at r =1 are A(1; t) = 0; B(1; t) = B ; C(1; t) = 0 ; (1.7) where B is a positive constant. We now impose initial conditions. First, R(0) = R0 > 0 : (1.8)