COAGULATION PROCESSES WITH MASS-LOSS

Smoluchowski's coagulation equation, with reaction rate K(x, y), descr ibing the time evolution of a size distribution c(x, t) is studied in the presence of a mass loss term m(x) = mx(m > 0). For K(x, y) = 1, c( x, t) is determined explicitly for arbitrary initial distributions. If K(x, y) = xy, we determine c(x, t) explicitly for arbitrary initial d istributions and describe the behaviour of c(x, t) for large x, for al l times. Here, we show that a phase transition occurs in a finite time t(g) = -(1/2m) log(e)(1 - 2m) provided m < 1/2. An investigation into K(x, y) = (xy)(omega) reveals that a phase transition occurs in a fin ite time t(g) if and only if 1/2 < omega less than or equal to 1 and m < 1/2. An estimate of the least upper bound for t(g) is calculated, a nd the behaviour of c(x, t) for large x with t > t(g) is presented.