  We present a generalized model of a diffusion-reaction system where the
reaction occurs only on the boundary. This model reduces to that of Barato and
Hinrichsen when the occupancy of the boundary site is restricted to zero or
one. In the limit when there is no restriction on the occupancy of the boundary
site, the model reduces to an age dependent Galton-Watson branching process and
admits an analytic solution. The model displays a boundary-induced phase
transition into an absorbing state with rational critical exponents and
exhibits aging at criticality below a certain fractal dimension of the
diffusion process. Surprisingly the behavior in the critical regime for
intermediate occupancy restriction $N$ varies with $N$. In fact, by varying the
lifetime of the active boundary particle or the diffusion coefficient in the
bulk, the critical exponents can be continuously modified.
