The steady, spherically symmetric solutions to the reaction-diffusion equations based on a simple autocatalytic reaction followed by the decay of the autocatalyst are considered. Three parameters—the orders with respect to the autocatalyst in the autocatalysis p and in the decay q and the rate of decay of the autocatalyst relative to its autocatalytic production $K$—determine the steady concentration profiles. Numerical integrations for a fixed value of the order of the autocatalyst show that the concentration profiles have different forms depending on whether $q<p$ or $q>~p.\mathrm{}$ In the former case, there is a critical decay rate $K_{\mathrm{crit}}$ for solutions to exist, with multiple solutions for $K<\mathrm{}{K}_{\mathrm{crit}}.$ In the latter case, there is a single solution for each value of K. This difference in the nature of the solution is confirmed by an analysis for p large. The temporal stability of the isothermal flame balls is examined, with temporally stable solutions being possible, provided that the ratio of the diffusion coefficient of the autocatalyst to that of the reactant is sufficiently small. The change in stability appears only when there are multiple solutions and is through a subcritical Hopf bifurcation.

• Received 12 May 2003

DOI:https://doi.org/10.1103/PhysRevE.68.036210