Large-time behavior of the generalized Smoluchovski coagulation equations

摘要

We study the large-time asymptotic solutions of the generalized Smoluchovski equations for class I and class II coagulation systems. It is found that, in gelling and nongelling systems of class I, the general solution ck(t) approaches for t--> [infinity] the exact solution Cbk/t (k finite), where the bk are independent of the initial conditions ck(0) and can be determined from a recursion relation. In class II systems, if the k and t dependence of ck(t) factorizes for large time, i.e., ck(t)-->c1(t)bk (t--> [infinity] ,k finite), then the bk (~k- tau ) can be obtained from a recursion relation. We show that if ck(t) is factorizable at large time, then the scaling function method and the recursion relation method give the same result for the tau exponent for the class II systems.