THE DEVELOPMENT OF TRAVELING WAVES IN QUADRATIC AND CUBIC AUTOCATALYSIS WITH UNEQUAL DIFFUSION RATES .2. AN INITIAL-VALUE PROBLEM WITH AN IMMOBILIZED OR NEARLY IMMOBILIZED AUTOCATALYST

We study the isothermal autocatalytic reaction schemes, A + B --> 2B (quadratic autocatalysis), and A + 2B --> 3B (cubic autocatalysis), where A is a reactant and B is an autocatalyst. We consider the situation when a quantity of B is introduced locally into a uniform expanse of A, in one-dimensional slab geometry. In addition, we allow the chemical species A and B to have unequal diffusion rates D(A) and D(B), respectively, and study the two closely related cases, (D(B)/D(A)) = 0 and 0 < (D(B)/D(A)) << 1. When (D(B)/D(A)) = 0 a spike forms in the concentration of B, which grows indefinitely, and we can obtain both large and small time asymptotic solutions. For 0 < (D(B)/D(A)) << 1 there is a long induction period during which a large spike forms in the concentration of B, before a minimum speed travelling wave is generated. We can relate the results for case (D(B)/D(A)) = 0 to the solution when 0 < (D(B)/D(A)) << 1 to obtain detailed information about its behaviour.