On the Reactions A+A+...+A->0 at a One-Dimensional Periodic Lattice of   Catalytic Centers: Exact Solution
  The kinetics of the diffusion-controlled chemical reactions A+A+...+A->0 that occur at catalytic centers periodically arranged along a straight line is considered. Modes of the behavior of reaction probability W(t) were studied at different times and different densities of the catalyst. Within the Smoluchowski approximation, it was rigorously proved that at large times the function W(t) is independent of the lattice period. This means that, in the given asymptotic mode, the probability of the reaction on a lattice with a catalyst placed in each lattice site is the same as on a lattice with a catalyst placed in sparse sites. 
A.A.Naidenov and S.K.Nechaev