Singularities in the kinetics of coagulation processes

The authors consider a system of substances A_i reacting according to the following scheme: A_k+A_l-R_Kl to _k+l, (R_kl=R_lk>or=0). (The reaction is taken as irreversible.) They discuss the existence of global solutions of the kinetic equations derived for the concentrations. It is shown that one cannot expect the total number of monomers to remain constant. Rather, it can decrease as the result of the formation of infinite clusters (gelation). With this restriction, they obtain that a physically reasonable global solution exists if R_kl<or=r_kr_l and r_k=o(k). It is further conjectured that no gelation will take place if r_k=o( square root k). The case R_jk=(Aj+B)(Ak+B) is also solved explicitly and shown to exhibit gelation at t=1/A(A+B).