A boundary-driven reaction front

A model reaction scheme in which two species $A$ and $B$ react to form an inert product is considered, with the possible linear decay of $A$ to a further inert prduct also included. The reaction between $A$ and $B$ is maintained by the input of $A$ from the boundary which keeps $A$ at a constant concentration. The cases when $B$ is immobile or free to diffuse are treated. In the former case reaction fronts in $B$ are seen to develop. Large time asymptotic solutions are derived which show that these fronts propagate across the reactor at rates proportional to $t^{1/2}$ or $\log t$ ( $t$ is a dimensionless time) depending on whether the extra decay step is included. A similar situation is seen when $B$ can diffuse when the linear decay step is not present. However, when this extra step is included in the reaction scheme the reaction zone reaches only a finite distance fronm the boundary at large times.